TSTP Solution File: KLE104-10 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE104-10 : TPTP v8.1.2. Released v7.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:35:54 EDT 2023
% Result : Unsatisfiable 182.72s 23.60s
% Output : Proof 191.27s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : KLE104-10 : TPTP v8.1.2. Released v7.3.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n018.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 12:32:47 EDT 2023
% 0.13/0.35 % CPUTime :
% 182.72/23.60 Command-line arguments: --no-flatten-goal
% 182.72/23.60
% 182.72/23.60 % SZS status Unsatisfiable
% 182.72/23.60
% 189.89/24.57 % SZS output start Proof
% 189.89/24.57 Axiom 1 (codomain4): codomain(X) = coantidomain(coantidomain(X)).
% 189.89/24.57 Axiom 2 (complement): c(X) = antidomain(domain(X)).
% 189.89/24.57 Axiom 3 (domain4): domain(X) = antidomain(antidomain(X)).
% 189.89/24.57 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 189.89/24.57 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 189.89/24.57 Axiom 6 (additive_identity): addition(X, zero) = X.
% 189.89/24.57 Axiom 7 (multiplicative_right_identity): multiplication(X, one) = X.
% 189.89/24.57 Axiom 8 (right_annihilation): multiplication(X, zero) = zero.
% 189.89/24.57 Axiom 9 (multiplicative_left_identity): multiplication(one, X) = X.
% 189.89/24.57 Axiom 10 (left_annihilation): multiplication(zero, X) = zero.
% 189.89/24.57 Axiom 11 (codomain1): multiplication(X, coantidomain(X)) = zero.
% 189.89/24.57 Axiom 12 (domain1): multiplication(antidomain(X), X) = zero.
% 189.89/24.57 Axiom 13 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 189.89/24.57 Axiom 14 (ifeq_axiom_001): ifeq(X, X, Y, Z) = Y.
% 189.89/24.57 Axiom 15 (backward_diamond): backward_diamond(X, Y) = codomain(multiplication(codomain(Y), X)).
% 189.89/24.57 Axiom 16 (forward_box): forward_box(X, Y) = c(forward_diamond(X, c(Y))).
% 189.89/24.57 Axiom 17 (backward_box): backward_box(X, Y) = c(backward_diamond(X, c(Y))).
% 189.89/24.57 Axiom 18 (forward_diamond): forward_diamond(X, Y) = domain(multiplication(X, domain(Y))).
% 189.89/24.57 Axiom 19 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 189.89/24.57 Axiom 20 (domain_difference): domain_difference(X, Y) = multiplication(domain(X), antidomain(Y)).
% 189.89/24.57 Axiom 21 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 189.89/24.57 Axiom 22 (codomain3): addition(coantidomain(coantidomain(X)), coantidomain(X)) = one.
% 189.89/24.57 Axiom 23 (domain3): addition(antidomain(antidomain(X)), antidomain(X)) = one.
% 189.89/24.57 Axiom 24 (goals): addition(domain(sK2_goals_X1), backward_box(sK3_goals_X0, domain(sK1_goals_X2))) = one.
% 189.89/24.57 Axiom 25 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 189.89/24.57 Axiom 26 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 189.89/24.57 Axiom 27 (order): ifeq2(addition(X, Y), Y, leq(X, Y), true) = true.
% 189.89/24.57 Axiom 28 (order_1): ifeq(leq(X, Y), true, addition(X, Y), Y) = Y.
% 189.89/24.57 Axiom 29 (codomain2): addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))) = coantidomain(multiplication(coantidomain(coantidomain(X)), Y)).
% 189.89/24.57 Axiom 30 (domain2): addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))) = antidomain(multiplication(X, antidomain(antidomain(Y)))).
% 189.89/24.57
% 189.89/24.57 Lemma 31: domain(antidomain(X)) = c(X).
% 189.89/24.57 Proof:
% 189.89/24.57 domain(antidomain(X))
% 189.89/24.57 = { by axiom 3 (domain4) }
% 189.89/24.57 antidomain(antidomain(antidomain(X)))
% 189.89/24.57 = { by axiom 3 (domain4) R->L }
% 189.89/24.57 antidomain(domain(X))
% 189.89/24.57 = { by axiom 2 (complement) R->L }
% 189.89/24.57 c(X)
% 189.89/24.57
% 189.89/24.57 Lemma 32: multiplication(addition(X, antidomain(Y)), Y) = multiplication(X, Y).
% 189.89/24.57 Proof:
% 189.89/24.57 multiplication(addition(X, antidomain(Y)), Y)
% 189.89/24.57 = { by axiom 26 (left_distributivity) }
% 189.89/24.57 addition(multiplication(X, Y), multiplication(antidomain(Y), Y))
% 189.89/24.57 = { by axiom 12 (domain1) }
% 189.89/24.57 addition(multiplication(X, Y), zero)
% 189.89/24.57 = { by axiom 6 (additive_identity) }
% 189.89/24.57 multiplication(X, Y)
% 189.89/24.57
% 189.89/24.57 Lemma 33: addition(domain(X), antidomain(X)) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 addition(domain(X), antidomain(X))
% 189.89/24.57 = { by axiom 3 (domain4) }
% 189.89/24.57 addition(antidomain(antidomain(X)), antidomain(X))
% 189.89/24.57 = { by axiom 23 (domain3) }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 34: multiplication(domain(X), X) = X.
% 189.89/24.57 Proof:
% 189.89/24.57 multiplication(domain(X), X)
% 189.89/24.57 = { by lemma 32 R->L }
% 189.89/24.57 multiplication(addition(domain(X), antidomain(X)), X)
% 189.89/24.57 = { by lemma 33 }
% 189.89/24.57 multiplication(one, X)
% 189.89/24.57 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.57 X
% 189.89/24.57
% 189.89/24.57 Lemma 35: multiplication(c(X), antidomain(X)) = antidomain(X).
% 189.89/24.57 Proof:
% 189.89/24.57 multiplication(c(X), antidomain(X))
% 189.89/24.57 = { by lemma 31 R->L }
% 189.89/24.57 multiplication(domain(antidomain(X)), antidomain(X))
% 189.89/24.57 = { by lemma 34 }
% 189.89/24.57 antidomain(X)
% 189.89/24.57
% 189.89/24.57 Lemma 36: multiplication(antidomain(X), addition(X, Y)) = multiplication(antidomain(X), Y).
% 189.89/24.57 Proof:
% 189.89/24.57 multiplication(antidomain(X), addition(X, Y))
% 189.89/24.57 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.57 multiplication(antidomain(X), addition(Y, X))
% 189.89/24.57 = { by axiom 25 (right_distributivity) }
% 189.89/24.57 addition(multiplication(antidomain(X), Y), multiplication(antidomain(X), X))
% 189.89/24.57 = { by axiom 12 (domain1) }
% 189.89/24.57 addition(multiplication(antidomain(X), Y), zero)
% 189.89/24.57 = { by axiom 6 (additive_identity) }
% 189.89/24.57 multiplication(antidomain(X), Y)
% 189.89/24.57
% 189.89/24.57 Lemma 37: antidomain(X) = c(X).
% 189.89/24.57 Proof:
% 189.89/24.57 antidomain(X)
% 189.89/24.57 = { by lemma 35 R->L }
% 189.89/24.57 multiplication(c(X), antidomain(X))
% 189.89/24.57 = { by axiom 2 (complement) }
% 189.89/24.57 multiplication(antidomain(domain(X)), antidomain(X))
% 189.89/24.57 = { by lemma 36 R->L }
% 189.89/24.57 multiplication(antidomain(domain(X)), addition(domain(X), antidomain(X)))
% 189.89/24.57 = { by lemma 33 }
% 189.89/24.57 multiplication(antidomain(domain(X)), one)
% 189.89/24.57 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.57 antidomain(domain(X))
% 189.89/24.57 = { by axiom 2 (complement) R->L }
% 189.89/24.57 c(X)
% 189.89/24.57
% 189.89/24.57 Lemma 38: antidomain(one) = zero.
% 189.89/24.57 Proof:
% 189.89/24.57 antidomain(one)
% 189.89/24.57 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.57 multiplication(antidomain(one), one)
% 189.89/24.57 = { by axiom 12 (domain1) }
% 189.89/24.57 zero
% 189.89/24.57
% 189.89/24.57 Lemma 39: domain(one) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 domain(one)
% 189.89/24.57 = { by axiom 6 (additive_identity) R->L }
% 189.89/24.57 addition(domain(one), zero)
% 189.89/24.57 = { by lemma 38 R->L }
% 189.89/24.57 addition(domain(one), antidomain(one))
% 189.89/24.57 = { by lemma 33 }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 40: antidomain(zero) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 antidomain(zero)
% 189.89/24.57 = { by lemma 38 R->L }
% 189.89/24.57 antidomain(antidomain(one))
% 189.89/24.57 = { by axiom 3 (domain4) R->L }
% 189.89/24.57 domain(one)
% 189.89/24.57 = { by lemma 39 }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 41: coantidomain(one) = zero.
% 189.89/24.57 Proof:
% 189.89/24.57 coantidomain(one)
% 189.89/24.57 = { by axiom 9 (multiplicative_left_identity) R->L }
% 189.89/24.57 multiplication(one, coantidomain(one))
% 189.89/24.57 = { by axiom 11 (codomain1) }
% 189.89/24.57 zero
% 189.89/24.57
% 189.89/24.57 Lemma 42: addition(codomain(X), coantidomain(X)) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 addition(codomain(X), coantidomain(X))
% 189.89/24.57 = { by axiom 1 (codomain4) }
% 189.89/24.57 addition(coantidomain(coantidomain(X)), coantidomain(X))
% 189.89/24.57 = { by axiom 22 (codomain3) }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 43: codomain(one) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 codomain(one)
% 189.89/24.57 = { by axiom 6 (additive_identity) R->L }
% 189.89/24.57 addition(codomain(one), zero)
% 189.89/24.57 = { by lemma 41 R->L }
% 189.89/24.57 addition(codomain(one), coantidomain(one))
% 189.89/24.57 = { by lemma 42 }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 44: coantidomain(zero) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 coantidomain(zero)
% 189.89/24.57 = { by lemma 41 R->L }
% 189.89/24.57 coantidomain(coantidomain(one))
% 189.89/24.57 = { by axiom 1 (codomain4) R->L }
% 189.89/24.57 codomain(one)
% 189.89/24.57 = { by lemma 43 }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 45: c(zero) = one.
% 189.89/24.57 Proof:
% 189.89/24.57 c(zero)
% 189.89/24.57 = { by lemma 31 R->L }
% 189.89/24.57 domain(antidomain(zero))
% 189.89/24.57 = { by lemma 40 }
% 189.89/24.57 domain(one)
% 189.89/24.57 = { by lemma 39 }
% 189.89/24.57 one
% 189.89/24.57
% 189.89/24.57 Lemma 46: addition(zero, X) = X.
% 189.89/24.57 Proof:
% 189.89/24.57 addition(zero, X)
% 189.89/24.57 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.57 addition(X, zero)
% 189.89/24.57 = { by axiom 6 (additive_identity) }
% 189.89/24.57 X
% 189.89/24.57
% 189.89/24.57 Lemma 47: antidomain(c(X)) = domain(domain(X)).
% 189.89/24.57 Proof:
% 189.89/24.57 antidomain(c(X))
% 189.89/24.57 = { by axiom 2 (complement) }
% 189.89/24.57 antidomain(antidomain(domain(X)))
% 189.89/24.57 = { by axiom 3 (domain4) R->L }
% 189.89/24.57 domain(domain(X))
% 189.89/24.57
% 189.89/24.57 Lemma 48: antidomain(c(X)) = c(antidomain(X)).
% 189.89/24.57 Proof:
% 189.89/24.57 antidomain(c(X))
% 189.89/24.57 = { by lemma 31 R->L }
% 189.89/24.57 antidomain(domain(antidomain(X)))
% 189.89/24.57 = { by axiom 2 (complement) R->L }
% 189.89/24.57 c(antidomain(X))
% 189.89/24.57
% 189.89/24.57 Lemma 49: domain(domain(X)) = forward_diamond(one, X).
% 189.89/24.57 Proof:
% 189.89/24.57 domain(domain(X))
% 189.89/24.57 = { by axiom 9 (multiplicative_left_identity) R->L }
% 189.89/24.57 domain(multiplication(one, domain(X)))
% 189.89/24.57 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.58 forward_diamond(one, X)
% 189.89/24.58
% 189.89/24.58 Lemma 50: coantidomain(codomain(X)) = codomain(coantidomain(X)).
% 189.89/24.58 Proof:
% 189.89/24.58 coantidomain(codomain(X))
% 189.89/24.58 = { by axiom 1 (codomain4) }
% 189.89/24.58 coantidomain(coantidomain(coantidomain(X)))
% 189.89/24.58 = { by axiom 1 (codomain4) R->L }
% 189.89/24.58 codomain(coantidomain(X))
% 189.89/24.58
% 189.89/24.58 Lemma 51: multiplication(addition(X, Y), coantidomain(X)) = multiplication(Y, coantidomain(X)).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(addition(X, Y), coantidomain(X))
% 189.89/24.58 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.58 multiplication(addition(Y, X), coantidomain(X))
% 189.89/24.58 = { by axiom 26 (left_distributivity) }
% 189.89/24.58 addition(multiplication(Y, coantidomain(X)), multiplication(X, coantidomain(X)))
% 189.89/24.58 = { by axiom 11 (codomain1) }
% 189.89/24.58 addition(multiplication(Y, coantidomain(X)), zero)
% 189.89/24.58 = { by axiom 6 (additive_identity) }
% 189.89/24.58 multiplication(Y, coantidomain(X))
% 189.89/24.58
% 189.89/24.58 Lemma 52: multiplication(X, addition(Y, coantidomain(X))) = multiplication(X, Y).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(X, addition(Y, coantidomain(X)))
% 189.89/24.58 = { by axiom 25 (right_distributivity) }
% 189.89/24.58 addition(multiplication(X, Y), multiplication(X, coantidomain(X)))
% 189.89/24.58 = { by axiom 11 (codomain1) }
% 189.89/24.58 addition(multiplication(X, Y), zero)
% 189.89/24.58 = { by axiom 6 (additive_identity) }
% 189.89/24.58 multiplication(X, Y)
% 189.89/24.58
% 189.89/24.58 Lemma 53: multiplication(X, codomain(X)) = X.
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(X, codomain(X))
% 189.89/24.58 = { by lemma 52 R->L }
% 189.89/24.58 multiplication(X, addition(codomain(X), coantidomain(X)))
% 189.89/24.58 = { by lemma 42 }
% 189.89/24.58 multiplication(X, one)
% 189.89/24.58 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.58 X
% 189.89/24.58
% 189.89/24.58 Lemma 54: codomain(coantidomain(X)) = coantidomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 codomain(coantidomain(X))
% 189.89/24.58 = { by lemma 50 R->L }
% 189.89/24.58 coantidomain(codomain(X))
% 189.89/24.58 = { by axiom 9 (multiplicative_left_identity) R->L }
% 189.89/24.58 multiplication(one, coantidomain(codomain(X)))
% 189.89/24.58 = { by lemma 42 R->L }
% 189.89/24.58 multiplication(addition(codomain(X), coantidomain(X)), coantidomain(codomain(X)))
% 189.89/24.58 = { by lemma 51 }
% 189.89/24.58 multiplication(coantidomain(X), coantidomain(codomain(X)))
% 189.89/24.58 = { by lemma 50 }
% 189.89/24.58 multiplication(coantidomain(X), codomain(coantidomain(X)))
% 189.89/24.58 = { by lemma 53 }
% 189.89/24.58 coantidomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 55: coantidomain(codomain(X)) = coantidomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 coantidomain(codomain(X))
% 189.89/24.58 = { by axiom 1 (codomain4) }
% 189.89/24.58 coantidomain(coantidomain(coantidomain(X)))
% 189.89/24.58 = { by axiom 1 (codomain4) R->L }
% 189.89/24.58 codomain(coantidomain(X))
% 189.89/24.58 = { by lemma 54 }
% 189.89/24.58 coantidomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 56: multiplication(c(multiplication(X, Y)), multiplication(X, domain(Y))) = zero.
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(c(multiplication(X, Y)), multiplication(X, domain(Y)))
% 189.89/24.58 = { by lemma 37 R->L }
% 189.89/24.58 multiplication(antidomain(multiplication(X, Y)), multiplication(X, domain(Y)))
% 189.89/24.58 = { by lemma 32 R->L }
% 189.89/24.58 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y)))), multiplication(X, domain(Y)))
% 189.89/24.58 = { by axiom 3 (domain4) }
% 189.89/24.58 multiplication(addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y))))), multiplication(X, domain(Y)))
% 189.89/24.58 = { by axiom 30 (domain2) }
% 189.89/24.58 multiplication(antidomain(multiplication(X, antidomain(antidomain(Y)))), multiplication(X, domain(Y)))
% 189.89/24.58 = { by axiom 3 (domain4) R->L }
% 189.89/24.58 multiplication(antidomain(multiplication(X, domain(Y))), multiplication(X, domain(Y)))
% 189.89/24.58 = { by axiom 12 (domain1) }
% 189.89/24.58 zero
% 189.89/24.58
% 189.89/24.58 Lemma 57: addition(c(X), domain(X)) = one.
% 189.89/24.58 Proof:
% 189.89/24.58 addition(c(X), domain(X))
% 189.89/24.58 = { by axiom 3 (domain4) }
% 189.89/24.58 addition(c(X), antidomain(antidomain(X)))
% 189.89/24.58 = { by lemma 31 R->L }
% 189.89/24.58 addition(domain(antidomain(X)), antidomain(antidomain(X)))
% 189.89/24.58 = { by lemma 33 }
% 189.89/24.58 one
% 189.89/24.58
% 189.89/24.58 Lemma 58: multiplication(X, c(coantidomain(X))) = X.
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(X, c(coantidomain(X)))
% 189.89/24.58 = { by lemma 46 R->L }
% 189.89/24.58 addition(zero, multiplication(X, c(coantidomain(X))))
% 189.89/24.58 = { by lemma 56 R->L }
% 189.89/24.58 addition(multiplication(c(multiplication(X, coantidomain(X))), multiplication(X, domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 189.89/24.58 = { by axiom 11 (codomain1) }
% 189.89/24.58 addition(multiplication(c(zero), multiplication(X, domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 189.89/24.58 = { by lemma 45 }
% 189.89/24.58 addition(multiplication(one, multiplication(X, domain(coantidomain(X)))), multiplication(X, c(coantidomain(X))))
% 189.89/24.58 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.58 addition(multiplication(X, domain(coantidomain(X))), multiplication(X, c(coantidomain(X))))
% 189.89/24.58 = { by axiom 25 (right_distributivity) R->L }
% 189.89/24.58 multiplication(X, addition(domain(coantidomain(X)), c(coantidomain(X))))
% 189.89/24.58 = { by axiom 5 (additive_commutativity) }
% 189.89/24.58 multiplication(X, addition(c(coantidomain(X)), domain(coantidomain(X))))
% 189.89/24.58 = { by lemma 57 }
% 189.89/24.58 multiplication(X, one)
% 189.89/24.58 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.58 X
% 189.89/24.58
% 189.89/24.58 Lemma 59: addition(X, addition(X, Y)) = addition(X, Y).
% 189.89/24.58 Proof:
% 189.89/24.58 addition(X, addition(X, Y))
% 189.89/24.58 = { by axiom 19 (additive_associativity) }
% 189.89/24.58 addition(addition(X, X), Y)
% 189.89/24.58 = { by axiom 4 (additive_idempotence) }
% 189.89/24.58 addition(X, Y)
% 189.89/24.58
% 189.89/24.58 Lemma 60: multiplication(addition(X, one), Y) = addition(Y, multiplication(X, Y)).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(addition(X, one), Y)
% 189.89/24.58 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.58 multiplication(addition(one, X), Y)
% 189.89/24.58 = { by axiom 26 (left_distributivity) }
% 189.89/24.58 addition(multiplication(one, Y), multiplication(X, Y))
% 189.89/24.58 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.58 addition(Y, multiplication(X, Y))
% 189.89/24.58
% 189.89/24.58 Lemma 61: multiplication(addition(one, Y), X) = addition(X, multiplication(Y, X)).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(addition(one, Y), X)
% 189.89/24.58 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.58 multiplication(addition(Y, one), X)
% 189.89/24.58 = { by lemma 60 }
% 189.89/24.58 addition(X, multiplication(Y, X))
% 189.89/24.58
% 189.89/24.58 Lemma 62: leq(X, addition(Y, X)) = true.
% 189.89/24.58 Proof:
% 189.89/24.58 leq(X, addition(Y, X))
% 189.89/24.58 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.58 leq(X, addition(X, Y))
% 189.89/24.58 = { by axiom 13 (ifeq_axiom) R->L }
% 189.89/24.58 ifeq2(addition(X, Y), addition(X, Y), leq(X, addition(X, Y)), true)
% 189.89/24.58 = { by lemma 59 R->L }
% 189.89/24.58 ifeq2(addition(X, addition(X, Y)), addition(X, Y), leq(X, addition(X, Y)), true)
% 189.89/24.58 = { by axiom 27 (order) }
% 189.89/24.58 true
% 189.89/24.58
% 189.89/24.58 Lemma 63: backward_diamond(X, one) = codomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 backward_diamond(X, one)
% 189.89/24.58 = { by axiom 15 (backward_diamond) }
% 189.89/24.58 codomain(multiplication(codomain(one), X))
% 189.89/24.58 = { by lemma 43 }
% 189.89/24.58 codomain(multiplication(one, X))
% 189.89/24.58 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.58 codomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 64: c(codomain(X)) = backward_box(X, zero).
% 189.89/24.58 Proof:
% 189.89/24.58 c(codomain(X))
% 189.89/24.58 = { by lemma 63 R->L }
% 189.89/24.58 c(backward_diamond(X, one))
% 189.89/24.58 = { by lemma 45 R->L }
% 189.89/24.58 c(backward_diamond(X, c(zero)))
% 189.89/24.58 = { by axiom 17 (backward_box) R->L }
% 189.89/24.58 backward_box(X, zero)
% 189.89/24.58
% 189.89/24.58 Lemma 65: addition(one, domain(X)) = one.
% 189.89/24.58 Proof:
% 189.89/24.58 addition(one, domain(X))
% 189.89/24.58 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.58 addition(domain(X), one)
% 189.89/24.58 = { by lemma 33 R->L }
% 189.89/24.58 addition(domain(X), addition(domain(X), antidomain(X)))
% 189.89/24.58 = { by lemma 59 }
% 189.89/24.58 addition(domain(X), antidomain(X))
% 189.89/24.58 = { by lemma 33 }
% 189.89/24.58 one
% 189.89/24.58
% 189.89/24.58 Lemma 66: addition(one, c(X)) = one.
% 189.89/24.58 Proof:
% 189.89/24.58 addition(one, c(X))
% 189.89/24.58 = { by lemma 31 R->L }
% 189.89/24.58 addition(one, domain(antidomain(X)))
% 189.89/24.58 = { by lemma 65 }
% 189.89/24.58 one
% 189.89/24.58
% 189.89/24.58 Lemma 67: c(coantidomain(X)) = codomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 c(coantidomain(X))
% 189.89/24.58 = { by axiom 28 (order_1) R->L }
% 189.89/24.58 ifeq(leq(codomain(X), c(coantidomain(X))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 55 R->L }
% 189.89/24.58 ifeq(leq(codomain(X), c(coantidomain(codomain(X)))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 58 R->L }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), c(coantidomain(codomain(X)))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by axiom 9 (multiplicative_left_identity) R->L }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(one, c(coantidomain(codomain(X))))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 42 R->L }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(codomain(X), coantidomain(X)), c(coantidomain(codomain(X))))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 59 R->L }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(codomain(X), addition(codomain(X), coantidomain(X))), c(coantidomain(codomain(X))))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 42 }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(codomain(X), one), c(coantidomain(codomain(X))))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by axiom 5 (additive_commutativity) }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), multiplication(addition(one, codomain(X)), c(coantidomain(codomain(X))))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 61 }
% 189.89/24.58 ifeq(leq(multiplication(codomain(X), c(coantidomain(codomain(X)))), addition(c(coantidomain(codomain(X))), multiplication(codomain(X), c(coantidomain(codomain(X)))))), true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 62 }
% 189.89/24.58 ifeq(true, true, addition(codomain(X), c(coantidomain(X))), c(coantidomain(X)))
% 189.89/24.58 = { by axiom 14 (ifeq_axiom_001) }
% 189.89/24.58 addition(codomain(X), c(coantidomain(X)))
% 189.89/24.58 = { by lemma 54 R->L }
% 189.89/24.58 addition(codomain(X), c(codomain(coantidomain(X))))
% 189.89/24.58 = { by lemma 64 }
% 189.89/24.58 addition(codomain(X), backward_box(coantidomain(X), zero))
% 189.89/24.58 = { by axiom 1 (codomain4) }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), backward_box(coantidomain(X), zero))
% 189.89/24.58 = { by lemma 64 R->L }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), c(codomain(coantidomain(X))))
% 189.89/24.58 = { by lemma 37 R->L }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), antidomain(codomain(coantidomain(X))))
% 189.89/24.58 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), multiplication(antidomain(codomain(coantidomain(X))), one))
% 189.89/24.58 = { by lemma 42 R->L }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), multiplication(antidomain(codomain(coantidomain(X))), addition(codomain(coantidomain(X)), coantidomain(coantidomain(X)))))
% 189.89/24.58 = { by lemma 36 }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), multiplication(antidomain(codomain(coantidomain(X))), coantidomain(coantidomain(X))))
% 189.89/24.58 = { by lemma 37 }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), multiplication(c(codomain(coantidomain(X))), coantidomain(coantidomain(X))))
% 189.89/24.58 = { by lemma 64 }
% 189.89/24.58 addition(coantidomain(coantidomain(X)), multiplication(backward_box(coantidomain(X), zero), coantidomain(coantidomain(X))))
% 189.89/24.58 = { by lemma 61 R->L }
% 189.89/24.58 multiplication(addition(one, backward_box(coantidomain(X), zero)), coantidomain(coantidomain(X)))
% 189.89/24.58 = { by axiom 17 (backward_box) }
% 189.89/24.58 multiplication(addition(one, c(backward_diamond(coantidomain(X), c(zero)))), coantidomain(coantidomain(X)))
% 189.89/24.58 = { by lemma 66 }
% 189.89/24.58 multiplication(one, coantidomain(coantidomain(X)))
% 189.89/24.58 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.58 coantidomain(coantidomain(X))
% 189.89/24.58 = { by axiom 1 (codomain4) R->L }
% 189.89/24.58 codomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 68: domain(c(X)) = c(X).
% 189.89/24.58 Proof:
% 189.89/24.58 domain(c(X))
% 189.89/24.58 = { by lemma 37 R->L }
% 189.89/24.58 domain(antidomain(X))
% 189.89/24.58 = { by lemma 31 }
% 189.89/24.58 c(X)
% 189.89/24.58
% 189.89/24.58 Lemma 69: domain(codomain(X)) = codomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 domain(codomain(X))
% 189.89/24.58 = { by lemma 67 R->L }
% 189.89/24.58 domain(c(coantidomain(X)))
% 189.89/24.58 = { by lemma 68 }
% 189.89/24.58 c(coantidomain(X))
% 189.89/24.58 = { by lemma 67 }
% 189.89/24.58 codomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 70: backward_box(X, zero) = coantidomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 backward_box(X, zero)
% 189.89/24.58 = { by lemma 64 R->L }
% 189.89/24.58 c(codomain(X))
% 189.89/24.58 = { by axiom 1 (codomain4) }
% 189.89/24.58 c(coantidomain(coantidomain(X)))
% 189.89/24.58 = { by lemma 67 }
% 189.89/24.58 codomain(coantidomain(X))
% 189.89/24.58 = { by lemma 54 }
% 189.89/24.58 coantidomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 71: codomain(codomain(X)) = backward_diamond(one, X).
% 189.89/24.58 Proof:
% 189.89/24.58 codomain(codomain(X))
% 189.89/24.58 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.58 codomain(multiplication(codomain(X), one))
% 189.89/24.58 = { by axiom 15 (backward_diamond) R->L }
% 189.89/24.58 backward_diamond(one, X)
% 189.89/24.58
% 189.89/24.58 Lemma 72: backward_diamond(one, X) = codomain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 backward_diamond(one, X)
% 189.89/24.58 = { by lemma 71 R->L }
% 189.89/24.58 codomain(codomain(X))
% 189.89/24.58 = { by axiom 1 (codomain4) }
% 189.89/24.58 codomain(coantidomain(coantidomain(X)))
% 189.89/24.58 = { by lemma 54 }
% 189.89/24.58 coantidomain(coantidomain(X))
% 189.89/24.58 = { by axiom 1 (codomain4) R->L }
% 189.89/24.58 codomain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 73: backward_box(c(X), zero) = backward_box(one, X).
% 189.89/24.58 Proof:
% 189.89/24.58 backward_box(c(X), zero)
% 189.89/24.58 = { by lemma 64 R->L }
% 189.89/24.58 c(codomain(c(X)))
% 189.89/24.58 = { by lemma 72 R->L }
% 189.89/24.58 c(backward_diamond(one, c(X)))
% 189.89/24.58 = { by axiom 17 (backward_box) R->L }
% 189.89/24.58 backward_box(one, X)
% 189.89/24.58
% 189.89/24.58 Lemma 74: coantidomain(c(X)) = backward_box(one, X).
% 189.89/24.58 Proof:
% 189.89/24.58 coantidomain(c(X))
% 189.89/24.58 = { by lemma 70 R->L }
% 189.89/24.58 backward_box(c(X), zero)
% 189.89/24.58 = { by lemma 73 }
% 189.89/24.58 backward_box(one, X)
% 189.89/24.58
% 189.89/24.58 Lemma 75: multiplication(domain(X), domain(Y)) = domain_difference(X, c(Y)).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(domain(X), domain(Y))
% 189.89/24.58 = { by axiom 3 (domain4) }
% 189.89/24.58 multiplication(domain(X), antidomain(antidomain(Y)))
% 189.89/24.58 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.58 domain_difference(X, antidomain(Y))
% 189.89/24.58 = { by lemma 37 }
% 189.89/24.58 domain_difference(X, c(Y))
% 189.89/24.58
% 189.89/24.58 Lemma 76: addition(domain_difference(X, Y), domain_difference(X, c(Y))) = domain(X).
% 189.89/24.58 Proof:
% 189.89/24.58 addition(domain_difference(X, Y), domain_difference(X, c(Y)))
% 189.89/24.58 = { by lemma 75 R->L }
% 189.89/24.58 addition(domain_difference(X, Y), multiplication(domain(X), domain(Y)))
% 189.89/24.58 = { by axiom 20 (domain_difference) }
% 189.89/24.58 addition(multiplication(domain(X), antidomain(Y)), multiplication(domain(X), domain(Y)))
% 189.89/24.58 = { by axiom 25 (right_distributivity) R->L }
% 189.89/24.58 multiplication(domain(X), addition(antidomain(Y), domain(Y)))
% 189.89/24.58 = { by lemma 37 }
% 189.89/24.58 multiplication(domain(X), addition(c(Y), domain(Y)))
% 189.89/24.58 = { by lemma 57 }
% 189.89/24.58 multiplication(domain(X), one)
% 189.89/24.58 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.58 domain(X)
% 189.89/24.58
% 189.89/24.58 Lemma 77: c(antidomain(X)) = forward_diamond(one, X).
% 189.89/24.58 Proof:
% 189.89/24.58 c(antidomain(X))
% 189.89/24.58 = { by lemma 31 R->L }
% 189.89/24.58 domain(antidomain(antidomain(X)))
% 189.89/24.58 = { by axiom 3 (domain4) R->L }
% 189.89/24.58 domain(domain(X))
% 189.89/24.58 = { by lemma 49 }
% 189.89/24.58 forward_diamond(one, X)
% 189.89/24.58
% 189.89/24.58 Lemma 78: multiplication(domain(X), domain(Y)) = domain_difference(X, antidomain(Y)).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(domain(X), domain(Y))
% 189.89/24.58 = { by axiom 3 (domain4) }
% 189.89/24.58 multiplication(domain(X), antidomain(antidomain(Y)))
% 189.89/24.58 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.58 domain_difference(X, antidomain(Y))
% 189.89/24.58
% 189.89/24.58 Lemma 79: multiplication(c(X), domain(Y)) = domain_difference(antidomain(X), antidomain(Y)).
% 189.89/24.58 Proof:
% 189.89/24.58 multiplication(c(X), domain(Y))
% 189.89/24.58 = { by lemma 31 R->L }
% 189.89/24.58 multiplication(domain(antidomain(X)), domain(Y))
% 189.89/24.58 = { by lemma 78 }
% 189.89/24.58 domain_difference(antidomain(X), antidomain(Y))
% 189.89/24.59
% 189.89/24.59 Lemma 80: domain_difference(antidomain(X), X) = antidomain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 domain_difference(antidomain(X), X)
% 189.89/24.59 = { by axiom 20 (domain_difference) }
% 189.89/24.59 multiplication(domain(antidomain(X)), antidomain(X))
% 189.89/24.59 = { by lemma 34 }
% 189.89/24.59 antidomain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 81: forward_diamond(one, X) = domain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 forward_diamond(one, X)
% 189.89/24.59 = { by lemma 77 R->L }
% 189.89/24.59 c(antidomain(X))
% 189.89/24.59 = { by lemma 48 R->L }
% 189.89/24.59 antidomain(c(X))
% 189.89/24.59 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.59 multiplication(antidomain(c(X)), one)
% 189.89/24.59 = { by lemma 57 R->L }
% 189.89/24.59 multiplication(antidomain(c(X)), addition(c(X), domain(X)))
% 189.89/24.59 = { by lemma 36 }
% 189.89/24.59 multiplication(antidomain(c(X)), domain(X))
% 189.89/24.59 = { by lemma 48 }
% 189.89/24.59 multiplication(c(antidomain(X)), domain(X))
% 189.89/24.59 = { by lemma 79 }
% 189.89/24.59 domain_difference(antidomain(antidomain(X)), antidomain(X))
% 189.89/24.59 = { by lemma 80 }
% 189.89/24.59 antidomain(antidomain(X))
% 189.89/24.59 = { by axiom 3 (domain4) R->L }
% 189.89/24.59 domain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 82: multiplication(X, addition(Y, one)) = addition(X, multiplication(X, Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(X, addition(Y, one))
% 189.89/24.59 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.59 multiplication(X, addition(one, Y))
% 189.89/24.59 = { by axiom 25 (right_distributivity) }
% 189.89/24.59 addition(multiplication(X, one), multiplication(X, Y))
% 189.89/24.59 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.59 addition(X, multiplication(X, Y))
% 189.89/24.59
% 189.89/24.59 Lemma 83: addition(c(X), codomain(domain(X))) = one.
% 189.89/24.59 Proof:
% 189.89/24.59 addition(c(X), codomain(domain(X)))
% 189.89/24.59 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.59 addition(codomain(domain(X)), c(X))
% 189.89/24.59 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.59 addition(codomain(domain(X)), multiplication(c(X), one))
% 189.89/24.59 = { by axiom 28 (order_1) R->L }
% 189.89/24.59 addition(codomain(domain(X)), multiplication(c(X), ifeq(leq(coantidomain(domain(X)), one), true, addition(coantidomain(domain(X)), one), one)))
% 189.89/24.59 = { by lemma 42 R->L }
% 189.89/24.59 addition(codomain(domain(X)), multiplication(c(X), ifeq(leq(coantidomain(domain(X)), addition(codomain(domain(X)), coantidomain(domain(X)))), true, addition(coantidomain(domain(X)), one), one)))
% 189.89/24.59 = { by lemma 62 }
% 189.89/24.59 addition(codomain(domain(X)), multiplication(c(X), ifeq(true, true, addition(coantidomain(domain(X)), one), one)))
% 189.89/24.59 = { by axiom 14 (ifeq_axiom_001) }
% 189.89/24.59 addition(codomain(domain(X)), multiplication(c(X), addition(coantidomain(domain(X)), one)))
% 189.89/24.59 = { by lemma 82 }
% 189.89/24.59 addition(codomain(domain(X)), addition(c(X), multiplication(c(X), coantidomain(domain(X)))))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 addition(codomain(domain(X)), addition(c(X), multiplication(antidomain(X), coantidomain(domain(X)))))
% 189.89/24.59 = { by lemma 51 R->L }
% 189.89/24.59 addition(codomain(domain(X)), addition(c(X), multiplication(addition(domain(X), antidomain(X)), coantidomain(domain(X)))))
% 189.89/24.59 = { by lemma 33 }
% 189.89/24.59 addition(codomain(domain(X)), addition(c(X), multiplication(one, coantidomain(domain(X)))))
% 189.89/24.59 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.59 addition(codomain(domain(X)), addition(c(X), coantidomain(domain(X))))
% 189.89/24.59 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.59 addition(codomain(domain(X)), addition(coantidomain(domain(X)), c(X)))
% 189.89/24.59 = { by axiom 19 (additive_associativity) }
% 189.89/24.59 addition(addition(codomain(domain(X)), coantidomain(domain(X))), c(X))
% 189.89/24.59 = { by lemma 42 }
% 189.89/24.59 addition(one, c(X))
% 189.89/24.59 = { by lemma 66 }
% 189.89/24.59 one
% 189.89/24.59
% 189.89/24.59 Lemma 84: multiplication(codomain(domain(X)), X) = X.
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(codomain(domain(X)), X)
% 189.89/24.59 = { by lemma 46 R->L }
% 189.89/24.59 addition(zero, multiplication(codomain(domain(X)), X))
% 189.89/24.59 = { by axiom 12 (domain1) R->L }
% 189.89/24.59 addition(multiplication(antidomain(X), X), multiplication(codomain(domain(X)), X))
% 189.89/24.59 = { by axiom 26 (left_distributivity) R->L }
% 189.89/24.59 multiplication(addition(antidomain(X), codomain(domain(X))), X)
% 189.89/24.59 = { by lemma 37 }
% 189.89/24.59 multiplication(addition(c(X), codomain(domain(X))), X)
% 189.89/24.59 = { by lemma 83 }
% 189.89/24.59 multiplication(one, X)
% 189.89/24.59 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.59 X
% 189.89/24.59
% 189.89/24.59 Lemma 85: multiplication(codomain(domain(X)), domain(X)) = domain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(codomain(domain(X)), domain(X))
% 189.89/24.59 = { by lemma 81 R->L }
% 189.89/24.59 multiplication(codomain(forward_diamond(one, X)), domain(X))
% 189.89/24.59 = { by lemma 49 R->L }
% 189.89/24.59 multiplication(codomain(domain(domain(X))), domain(X))
% 189.89/24.59 = { by lemma 84 }
% 189.89/24.59 domain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 86: multiplication(domain(X), multiplication(X, Y)) = multiplication(X, Y).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(domain(X), multiplication(X, Y))
% 189.89/24.59 = { by axiom 21 (multiplicative_associativity) }
% 189.89/24.59 multiplication(multiplication(domain(X), X), Y)
% 189.89/24.59 = { by lemma 34 }
% 189.89/24.59 multiplication(X, Y)
% 189.89/24.59
% 189.89/24.59 Lemma 87: c(c(X)) = domain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 c(c(X))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 c(antidomain(X))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 antidomain(antidomain(X))
% 189.89/24.59 = { by axiom 3 (domain4) R->L }
% 189.89/24.59 domain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 88: multiplication(domain(X), c(Y)) = domain_difference(X, domain(Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(domain(X), c(Y))
% 189.89/24.59 = { by axiom 2 (complement) }
% 189.89/24.59 multiplication(domain(X), antidomain(domain(Y)))
% 189.89/24.59 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.59 domain_difference(X, domain(Y))
% 189.89/24.59
% 189.89/24.59 Lemma 89: domain_difference(X, domain(Y)) = domain_difference(X, Y).
% 189.89/24.59 Proof:
% 189.89/24.59 domain_difference(X, domain(Y))
% 189.89/24.59 = { by lemma 88 R->L }
% 189.89/24.59 multiplication(domain(X), c(Y))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 multiplication(domain(X), antidomain(Y))
% 189.89/24.59 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.59 domain_difference(X, Y)
% 189.89/24.59
% 189.89/24.59 Lemma 90: multiplication(c(X), c(Y)) = domain_difference(c(X), Y).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(c(X), c(Y))
% 189.89/24.59 = { by lemma 31 R->L }
% 189.89/24.59 multiplication(domain(antidomain(X)), c(Y))
% 189.89/24.59 = { by lemma 88 }
% 189.89/24.59 domain_difference(antidomain(X), domain(Y))
% 189.89/24.59 = { by lemma 89 }
% 189.89/24.59 domain_difference(antidomain(X), Y)
% 189.89/24.59 = { by lemma 37 }
% 189.89/24.59 domain_difference(c(X), Y)
% 189.89/24.59
% 189.89/24.59 Lemma 91: multiplication(codomain(X), c(Y)) = domain_difference(codomain(X), Y).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(codomain(X), c(Y))
% 189.89/24.59 = { by lemma 67 R->L }
% 189.89/24.59 multiplication(c(coantidomain(X)), c(Y))
% 189.89/24.59 = { by lemma 90 }
% 189.89/24.59 domain_difference(c(coantidomain(X)), Y)
% 189.89/24.59 = { by lemma 67 }
% 189.89/24.59 domain_difference(codomain(X), Y)
% 189.89/24.59
% 189.89/24.59 Lemma 92: multiplication(codomain(X), multiplication(Y, backward_diamond(Y, X))) = multiplication(codomain(X), Y).
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(codomain(X), multiplication(Y, backward_diamond(Y, X)))
% 189.89/24.59 = { by axiom 21 (multiplicative_associativity) }
% 189.89/24.59 multiplication(multiplication(codomain(X), Y), backward_diamond(Y, X))
% 189.89/24.59 = { by axiom 15 (backward_diamond) }
% 189.89/24.59 multiplication(multiplication(codomain(X), Y), codomain(multiplication(codomain(X), Y)))
% 189.89/24.59 = { by lemma 53 }
% 189.89/24.59 multiplication(codomain(X), Y)
% 189.89/24.59
% 189.89/24.59 Lemma 93: multiplication(coantidomain(multiplication(X, Y)), backward_diamond(Y, X)) = zero.
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(coantidomain(multiplication(X, Y)), backward_diamond(Y, X))
% 189.89/24.59 = { by axiom 15 (backward_diamond) }
% 189.89/24.59 multiplication(coantidomain(multiplication(X, Y)), codomain(multiplication(codomain(X), Y)))
% 189.89/24.59 = { by axiom 1 (codomain4) }
% 189.89/24.59 multiplication(coantidomain(multiplication(X, Y)), coantidomain(coantidomain(multiplication(codomain(X), Y))))
% 189.89/24.59 = { by lemma 51 R->L }
% 189.89/24.59 multiplication(addition(coantidomain(multiplication(codomain(X), Y)), coantidomain(multiplication(X, Y))), coantidomain(coantidomain(multiplication(codomain(X), Y))))
% 189.89/24.59 = { by axiom 5 (additive_commutativity) }
% 189.89/24.59 multiplication(addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(codomain(X), Y))), coantidomain(coantidomain(multiplication(codomain(X), Y))))
% 189.89/24.59 = { by axiom 1 (codomain4) }
% 189.89/24.59 multiplication(addition(coantidomain(multiplication(X, Y)), coantidomain(multiplication(coantidomain(coantidomain(X)), Y))), coantidomain(coantidomain(multiplication(codomain(X), Y))))
% 189.89/24.59 = { by axiom 29 (codomain2) }
% 189.89/24.59 multiplication(coantidomain(multiplication(coantidomain(coantidomain(X)), Y)), coantidomain(coantidomain(multiplication(codomain(X), Y))))
% 189.89/24.59 = { by axiom 1 (codomain4) R->L }
% 189.89/24.59 multiplication(coantidomain(multiplication(codomain(X), Y)), coantidomain(coantidomain(multiplication(codomain(X), Y))))
% 189.89/24.59 = { by axiom 11 (codomain1) }
% 189.89/24.59 zero
% 189.89/24.59
% 189.89/24.59 Lemma 94: codomain(domain(X)) = domain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 codomain(domain(X))
% 189.89/24.59 = { by lemma 69 R->L }
% 189.89/24.59 domain(codomain(domain(X)))
% 189.89/24.59 = { by lemma 76 R->L }
% 189.89/24.59 addition(domain_difference(codomain(domain(X)), c(X)), domain_difference(codomain(domain(X)), c(c(X))))
% 189.89/24.59 = { by lemma 75 R->L }
% 189.89/24.59 addition(multiplication(domain(codomain(domain(X))), domain(X)), domain_difference(codomain(domain(X)), c(c(X))))
% 189.89/24.59 = { by lemma 85 R->L }
% 189.89/24.59 addition(multiplication(domain(codomain(domain(X))), multiplication(codomain(domain(X)), domain(X))), domain_difference(codomain(domain(X)), c(c(X))))
% 189.89/24.59 = { by lemma 86 }
% 189.89/24.59 addition(multiplication(codomain(domain(X)), domain(X)), domain_difference(codomain(domain(X)), c(c(X))))
% 189.89/24.59 = { by lemma 85 }
% 189.89/24.59 addition(domain(X), domain_difference(codomain(domain(X)), c(c(X))))
% 189.89/24.59 = { by lemma 87 }
% 189.89/24.59 addition(domain(X), domain_difference(codomain(domain(X)), domain(X)))
% 189.89/24.59 = { by lemma 89 }
% 189.89/24.59 addition(domain(X), domain_difference(codomain(domain(X)), X))
% 189.89/24.59 = { by lemma 87 R->L }
% 189.89/24.59 addition(domain(X), domain_difference(codomain(c(c(X))), X))
% 189.89/24.59 = { by lemma 91 R->L }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), c(X)))
% 189.89/24.59 = { by lemma 92 R->L }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), multiplication(c(X), backward_diamond(c(X), c(c(X))))))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), multiplication(c(X), backward_diamond(c(X), antidomain(c(X))))))
% 189.89/24.59 = { by axiom 9 (multiplicative_left_identity) R->L }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), multiplication(c(X), multiplication(one, backward_diamond(c(X), antidomain(c(X)))))))
% 189.89/24.59 = { by lemma 44 R->L }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), multiplication(c(X), multiplication(coantidomain(zero), backward_diamond(c(X), antidomain(c(X)))))))
% 189.89/24.59 = { by axiom 12 (domain1) R->L }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), multiplication(c(X), multiplication(coantidomain(multiplication(antidomain(c(X)), c(X))), backward_diamond(c(X), antidomain(c(X)))))))
% 189.89/24.59 = { by lemma 93 }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), multiplication(c(X), zero)))
% 189.89/24.59 = { by axiom 8 (right_annihilation) }
% 189.89/24.59 addition(domain(X), multiplication(codomain(c(c(X))), zero))
% 189.89/24.59 = { by axiom 8 (right_annihilation) }
% 189.89/24.59 addition(domain(X), zero)
% 189.89/24.59 = { by axiom 6 (additive_identity) }
% 189.89/24.59 domain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 95: c(domain(X)) = c(X).
% 189.89/24.59 Proof:
% 189.89/24.59 c(domain(X))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 antidomain(domain(X))
% 189.89/24.59 = { by axiom 2 (complement) R->L }
% 189.89/24.59 c(X)
% 189.89/24.59
% 189.89/24.59 Lemma 96: c(codomain(X)) = coantidomain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 c(codomain(X))
% 189.89/24.59 = { by lemma 63 R->L }
% 189.89/24.59 c(backward_diamond(X, one))
% 189.89/24.59 = { by lemma 45 R->L }
% 189.89/24.59 c(backward_diamond(X, c(zero)))
% 189.89/24.59 = { by axiom 17 (backward_box) R->L }
% 189.89/24.59 backward_box(X, zero)
% 189.89/24.59 = { by lemma 70 }
% 189.89/24.59 coantidomain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 97: domain_difference(X, zero) = domain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 domain_difference(X, zero)
% 189.89/24.59 = { by axiom 20 (domain_difference) }
% 189.89/24.59 multiplication(domain(X), antidomain(zero))
% 189.89/24.59 = { by lemma 40 }
% 189.89/24.59 multiplication(domain(X), one)
% 189.89/24.59 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.59 domain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 98: forward_diamond(X, one) = domain(X).
% 189.89/24.59 Proof:
% 189.89/24.59 forward_diamond(X, one)
% 189.89/24.59 = { by axiom 18 (forward_diamond) }
% 189.89/24.59 domain(multiplication(X, domain(one)))
% 189.89/24.59 = { by lemma 39 }
% 189.89/24.59 domain(multiplication(X, one))
% 189.89/24.59 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.59 domain(X)
% 189.89/24.59
% 189.89/24.59 Lemma 99: forward_box(X, zero) = c(X).
% 189.89/24.59 Proof:
% 189.89/24.59 forward_box(X, zero)
% 189.89/24.59 = { by axiom 16 (forward_box) }
% 189.89/24.59 c(forward_diamond(X, c(zero)))
% 189.89/24.59 = { by lemma 45 }
% 189.89/24.59 c(forward_diamond(X, one))
% 189.89/24.59 = { by lemma 98 }
% 189.89/24.59 c(domain(X))
% 189.89/24.59 = { by lemma 95 }
% 189.89/24.59 c(X)
% 189.89/24.59
% 189.89/24.59 Lemma 100: domain(multiplication(X, c(Y))) = forward_diamond(X, antidomain(Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 domain(multiplication(X, c(Y)))
% 189.89/24.59 = { by lemma 31 R->L }
% 189.89/24.59 domain(multiplication(X, domain(antidomain(Y))))
% 189.89/24.59 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.59 forward_diamond(X, antidomain(Y))
% 189.89/24.59
% 189.89/24.59 Lemma 101: forward_diamond(domain(X), c(Y)) = domain(domain_difference(X, Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 forward_diamond(domain(X), c(Y))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 forward_diamond(domain(X), antidomain(Y))
% 189.89/24.59 = { by lemma 100 R->L }
% 189.89/24.59 domain(multiplication(domain(X), c(Y)))
% 189.89/24.59 = { by lemma 88 }
% 189.89/24.59 domain(domain_difference(X, domain(Y)))
% 189.89/24.59 = { by lemma 89 }
% 189.89/24.59 domain(domain_difference(X, Y))
% 189.89/24.59
% 189.89/24.59 Lemma 102: domain(domain_difference(c(X), Y)) = forward_diamond(c(X), c(Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 domain(domain_difference(c(X), Y))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 domain(domain_difference(antidomain(X), Y))
% 189.89/24.59 = { by lemma 101 R->L }
% 189.89/24.59 forward_diamond(domain(antidomain(X)), c(Y))
% 189.89/24.59 = { by lemma 31 }
% 189.89/24.59 forward_diamond(c(X), c(Y))
% 189.89/24.59
% 189.89/24.59 Lemma 103: domain(domain_difference(X, antidomain(Y))) = forward_diamond(domain(X), Y).
% 189.89/24.59 Proof:
% 189.89/24.59 domain(domain_difference(X, antidomain(Y)))
% 189.89/24.59 = { by lemma 78 R->L }
% 189.89/24.59 domain(multiplication(domain(X), domain(Y)))
% 189.89/24.59 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.59 forward_diamond(domain(X), Y)
% 189.89/24.59
% 189.89/24.59 Lemma 104: c(multiplication(X, domain(Y))) = antidomain(forward_diamond(X, Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 c(multiplication(X, domain(Y)))
% 189.89/24.59 = { by axiom 2 (complement) }
% 189.89/24.59 antidomain(domain(multiplication(X, domain(Y))))
% 189.89/24.59 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.59 antidomain(forward_diamond(X, Y))
% 189.89/24.59
% 189.89/24.59 Lemma 105: c(multiplication(X, c(Y))) = forward_box(X, Y).
% 189.89/24.59 Proof:
% 189.89/24.59 c(multiplication(X, c(Y)))
% 189.89/24.59 = { by lemma 31 R->L }
% 189.89/24.59 c(multiplication(X, domain(antidomain(Y))))
% 189.89/24.59 = { by lemma 104 }
% 189.89/24.59 antidomain(forward_diamond(X, antidomain(Y)))
% 189.89/24.59 = { by lemma 37 }
% 189.89/24.59 c(forward_diamond(X, antidomain(Y)))
% 189.89/24.59 = { by lemma 37 }
% 189.89/24.59 c(forward_diamond(X, c(Y)))
% 189.89/24.59 = { by axiom 16 (forward_box) R->L }
% 189.89/24.59 forward_box(X, Y)
% 189.89/24.59
% 189.89/24.59 Lemma 106: c(forward_box(X, Y)) = forward_diamond(X, c(Y)).
% 189.89/24.59 Proof:
% 189.89/24.59 c(forward_box(X, Y))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 antidomain(forward_box(X, Y))
% 189.89/24.59 = { by lemma 105 R->L }
% 189.89/24.59 antidomain(c(multiplication(X, c(Y))))
% 189.89/24.59 = { by lemma 47 }
% 189.89/24.59 domain(domain(multiplication(X, c(Y))))
% 189.89/24.59 = { by lemma 49 }
% 189.89/24.59 forward_diamond(one, multiplication(X, c(Y)))
% 189.89/24.59 = { by lemma 81 }
% 189.89/24.59 domain(multiplication(X, c(Y)))
% 189.89/24.59 = { by lemma 100 }
% 189.89/24.59 forward_diamond(X, antidomain(Y))
% 189.89/24.59 = { by lemma 37 }
% 189.89/24.59 forward_diamond(X, c(Y))
% 189.89/24.59
% 189.89/24.59 Lemma 107: addition(c(X), domain_difference(Y, X)) = c(X).
% 189.89/24.59 Proof:
% 189.89/24.59 addition(c(X), domain_difference(Y, X))
% 189.89/24.59 = { by lemma 37 R->L }
% 189.89/24.59 addition(antidomain(X), domain_difference(Y, X))
% 189.89/24.59 = { by axiom 20 (domain_difference) }
% 189.89/24.59 addition(antidomain(X), multiplication(domain(Y), antidomain(X)))
% 189.89/24.59 = { by lemma 61 R->L }
% 189.89/24.59 multiplication(addition(one, domain(Y)), antidomain(X))
% 189.89/24.59 = { by lemma 65 }
% 189.89/24.59 multiplication(one, antidomain(X))
% 189.89/24.59 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.59 antidomain(X)
% 189.89/24.59 = { by lemma 37 }
% 189.89/24.59 c(X)
% 189.89/24.59
% 189.89/24.59 Lemma 108: multiplication(domain(X), domain_difference(Y, X)) = zero.
% 189.89/24.59 Proof:
% 189.89/24.59 multiplication(domain(X), domain_difference(Y, X))
% 189.89/24.59 = { by lemma 81 R->L }
% 189.89/24.59 multiplication(forward_diamond(one, X), domain_difference(Y, X))
% 189.89/24.59 = { by lemma 77 R->L }
% 189.89/24.59 multiplication(c(antidomain(X)), domain_difference(Y, X))
% 189.89/24.59 = { by lemma 48 R->L }
% 189.89/24.59 multiplication(antidomain(c(X)), domain_difference(Y, X))
% 189.89/24.59 = { by lemma 36 R->L }
% 189.89/24.59 multiplication(antidomain(c(X)), addition(c(X), domain_difference(Y, X)))
% 189.89/24.59 = { by lemma 107 }
% 189.89/24.59 multiplication(antidomain(c(X)), c(X))
% 189.89/24.59 = { by axiom 12 (domain1) }
% 189.89/24.59 zero
% 189.89/24.59
% 189.89/24.59 Lemma 109: c(domain_difference(X, Y)) = forward_box(domain(X), Y).
% 189.89/24.59 Proof:
% 189.89/24.59 c(domain_difference(X, Y))
% 189.89/24.59 = { by lemma 89 R->L }
% 189.89/24.59 c(domain_difference(X, domain(Y)))
% 189.89/24.59 = { by lemma 88 R->L }
% 189.89/24.59 c(multiplication(domain(X), c(Y)))
% 189.89/24.59 = { by lemma 105 }
% 189.89/24.60 forward_box(domain(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 110: domain_difference(X, domain_difference(Y, X)) = domain(X).
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(X, domain_difference(Y, X))
% 189.89/24.60 = { by lemma 89 R->L }
% 189.89/24.60 domain_difference(X, domain_difference(Y, domain(X)))
% 189.89/24.60 = { by lemma 87 R->L }
% 189.89/24.60 domain_difference(X, domain_difference(Y, c(c(X))))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 domain_difference(X, domain_difference(Y, antidomain(c(X))))
% 189.89/24.60 = { by lemma 89 R->L }
% 189.89/24.60 domain_difference(X, domain(domain_difference(Y, antidomain(c(X)))))
% 189.89/24.60 = { by lemma 103 }
% 189.89/24.60 domain_difference(X, forward_diamond(domain(Y), c(X)))
% 189.89/24.60 = { by lemma 106 R->L }
% 189.89/24.60 domain_difference(X, c(forward_box(domain(Y), X)))
% 189.89/24.60 = { by lemma 46 R->L }
% 189.89/24.60 addition(zero, domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 56 R->L }
% 189.89/24.60 addition(multiplication(c(multiplication(domain(X), domain_difference(Y, X))), multiplication(domain(X), domain(domain_difference(Y, X)))), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 108 }
% 189.89/24.60 addition(multiplication(c(zero), multiplication(domain(X), domain(domain_difference(Y, X)))), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 45 }
% 189.89/24.60 addition(multiplication(one, multiplication(domain(X), domain(domain_difference(Y, X)))), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.60 addition(multiplication(domain(X), domain(domain_difference(Y, X))), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 78 }
% 189.89/24.60 addition(domain_difference(X, antidomain(domain_difference(Y, X))), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 addition(domain_difference(X, c(domain_difference(Y, X))), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 109 }
% 189.89/24.60 addition(domain_difference(X, forward_box(domain(Y), X)), domain_difference(X, c(forward_box(domain(Y), X))))
% 189.89/24.60 = { by lemma 76 }
% 189.89/24.60 domain(X)
% 189.89/24.60
% 189.89/24.60 Lemma 111: domain_difference(domain_difference(X, Y), Y) = domain(domain_difference(X, Y)).
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(domain_difference(X, Y), Y)
% 189.89/24.60 = { by lemma 89 R->L }
% 189.89/24.60 domain_difference(domain_difference(X, Y), domain(Y))
% 189.89/24.60 = { by lemma 110 R->L }
% 189.89/24.60 domain_difference(domain_difference(X, Y), domain_difference(Y, domain_difference(X, Y)))
% 189.89/24.60 = { by lemma 110 }
% 189.89/24.60 domain(domain_difference(X, Y))
% 189.89/24.60
% 189.89/24.60 Lemma 112: multiplication(forward_diamond(X, Y), c(Z)) = domain_difference(multiplication(X, domain(Y)), Z).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(forward_diamond(X, Y), c(Z))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 multiplication(forward_diamond(X, Y), antidomain(Z))
% 189.89/24.60 = { by axiom 18 (forward_diamond) }
% 189.89/24.60 multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 domain_difference(multiplication(X, domain(Y)), Z)
% 189.89/24.60
% 189.89/24.60 Lemma 113: domain(forward_diamond(X, Y)) = forward_diamond(X, Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain(forward_diamond(X, Y))
% 189.89/24.60 = { by lemma 81 R->L }
% 189.89/24.60 forward_diamond(one, forward_diamond(X, Y))
% 189.89/24.60 = { by lemma 77 R->L }
% 189.89/24.60 c(antidomain(forward_diamond(X, Y)))
% 189.89/24.60 = { by lemma 104 R->L }
% 189.89/24.60 c(c(multiplication(X, domain(Y))))
% 189.89/24.60 = { by lemma 87 }
% 189.89/24.60 domain(multiplication(X, domain(Y)))
% 189.89/24.60 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.60 forward_diamond(X, Y)
% 189.89/24.60
% 189.89/24.60 Lemma 114: domain_difference(multiplication(X, domain(Y)), Z) = domain_difference(forward_diamond(X, Y), Z).
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(multiplication(X, domain(Y)), Z)
% 189.89/24.60 = { by lemma 112 R->L }
% 189.89/24.60 multiplication(forward_diamond(X, Y), c(Z))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 multiplication(forward_diamond(X, Y), antidomain(Z))
% 189.89/24.60 = { by lemma 113 R->L }
% 189.89/24.60 multiplication(domain(forward_diamond(X, Y)), antidomain(Z))
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 domain_difference(forward_diamond(X, Y), Z)
% 189.89/24.60
% 189.89/24.60 Lemma 115: domain_difference(multiplication(X, c(Y)), Z) = domain_difference(forward_diamond(X, c(Y)), Z).
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(multiplication(X, c(Y)), Z)
% 189.89/24.60 = { by axiom 20 (domain_difference) }
% 189.89/24.60 multiplication(domain(multiplication(X, c(Y))), antidomain(Z))
% 189.89/24.60 = { by lemma 100 }
% 189.89/24.60 multiplication(forward_diamond(X, antidomain(Y)), antidomain(Z))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 multiplication(forward_diamond(X, c(Y)), antidomain(Z))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 multiplication(forward_diamond(X, c(Y)), c(Z))
% 189.89/24.60 = { by lemma 112 }
% 189.89/24.60 domain_difference(multiplication(X, domain(c(Y))), Z)
% 189.89/24.60 = { by lemma 114 }
% 189.89/24.60 domain_difference(forward_diamond(X, c(Y)), Z)
% 189.89/24.60
% 189.89/24.60 Lemma 116: multiplication(forward_diamond(X, Y), c(Z)) = domain_difference(forward_diamond(X, Y), Z).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(forward_diamond(X, Y), c(Z))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 multiplication(forward_diamond(X, Y), antidomain(Z))
% 189.89/24.60 = { by axiom 18 (forward_diamond) }
% 189.89/24.60 multiplication(domain(multiplication(X, domain(Y))), antidomain(Z))
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 domain_difference(multiplication(X, domain(Y)), Z)
% 189.89/24.60 = { by lemma 114 }
% 189.89/24.60 domain_difference(forward_diamond(X, Y), Z)
% 189.89/24.60
% 189.89/24.60 Lemma 117: multiplication(domain(X), multiplication(domain(Y), Z)) = multiplication(domain_difference(X, c(Y)), Z).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(domain(X), multiplication(domain(Y), Z))
% 189.89/24.60 = { by axiom 21 (multiplicative_associativity) }
% 189.89/24.60 multiplication(multiplication(domain(X), domain(Y)), Z)
% 189.89/24.60 = { by lemma 78 }
% 189.89/24.60 multiplication(domain_difference(X, antidomain(Y)), Z)
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 multiplication(domain_difference(X, c(Y)), Z)
% 189.89/24.60
% 189.89/24.60 Lemma 118: multiplication(domain_difference(X, c(Y)), Y) = multiplication(domain(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(domain_difference(X, c(Y)), Y)
% 189.89/24.60 = { by lemma 117 R->L }
% 189.89/24.60 multiplication(domain(X), multiplication(domain(Y), Y))
% 189.89/24.60 = { by lemma 34 }
% 189.89/24.60 multiplication(domain(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 119: multiplication(antidomain(X), multiplication(X, Y)) = zero.
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(antidomain(X), multiplication(X, Y))
% 189.89/24.60 = { by axiom 21 (multiplicative_associativity) }
% 189.89/24.60 multiplication(multiplication(antidomain(X), X), Y)
% 189.89/24.60 = { by axiom 12 (domain1) }
% 189.89/24.60 multiplication(zero, Y)
% 189.89/24.60 = { by axiom 10 (left_annihilation) }
% 189.89/24.60 zero
% 189.89/24.60
% 189.89/24.60 Lemma 120: domain_difference(c(X), multiplication(X, Y)) = c(X).
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(c(X), multiplication(X, Y))
% 189.89/24.60 = { by lemma 89 R->L }
% 189.89/24.60 domain_difference(c(X), domain(multiplication(X, Y)))
% 189.89/24.60 = { by lemma 87 R->L }
% 189.89/24.60 domain_difference(c(X), c(c(multiplication(X, Y))))
% 189.89/24.60 = { by lemma 46 R->L }
% 189.89/24.60 addition(zero, domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 56 R->L }
% 189.89/24.60 addition(multiplication(c(multiplication(antidomain(X), multiplication(X, Y))), multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 119 }
% 189.89/24.60 addition(multiplication(c(zero), multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 45 }
% 189.89/24.60 addition(multiplication(one, multiplication(antidomain(X), domain(multiplication(X, Y)))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by axiom 9 (multiplicative_left_identity) }
% 189.89/24.60 addition(multiplication(antidomain(X), domain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 addition(multiplication(c(X), domain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 79 }
% 189.89/24.60 addition(domain_difference(antidomain(X), antidomain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 addition(domain_difference(c(X), antidomain(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 addition(domain_difference(c(X), c(multiplication(X, Y))), domain_difference(c(X), c(c(multiplication(X, Y)))))
% 189.89/24.60 = { by lemma 76 }
% 189.89/24.60 domain(c(X))
% 189.89/24.60 = { by lemma 68 }
% 189.89/24.60 c(X)
% 189.89/24.60
% 189.89/24.60 Lemma 121: addition(c(X), c(multiplication(X, Y))) = c(multiplication(X, Y)).
% 189.89/24.60 Proof:
% 189.89/24.60 addition(c(X), c(multiplication(X, Y)))
% 189.89/24.60 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), c(X))
% 189.89/24.60 = { by lemma 120 R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), domain_difference(c(X), multiplication(X, Y)))
% 189.89/24.60 = { by lemma 107 }
% 189.89/24.60 c(multiplication(X, Y))
% 189.89/24.60
% 189.89/24.60 Lemma 122: c(forward_diamond(X, Y)) = forward_box(X, c(Y)).
% 189.89/24.60 Proof:
% 189.89/24.60 c(forward_diamond(X, Y))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 antidomain(forward_diamond(X, Y))
% 189.89/24.60 = { by lemma 104 R->L }
% 189.89/24.60 c(multiplication(X, domain(Y)))
% 189.89/24.60 = { by lemma 81 R->L }
% 189.89/24.60 c(multiplication(X, forward_diamond(one, Y)))
% 189.89/24.60 = { by lemma 77 R->L }
% 189.89/24.60 c(multiplication(X, c(antidomain(Y))))
% 189.89/24.60 = { by lemma 105 }
% 189.89/24.60 forward_box(X, antidomain(Y))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 forward_box(X, c(Y))
% 189.89/24.60
% 189.89/24.60 Lemma 123: addition(forward_box(X, c(Y)), c(multiplication(X, Y))) = forward_box(X, c(Y)).
% 189.89/24.60 Proof:
% 189.89/24.60 addition(forward_box(X, c(Y)), c(multiplication(X, Y)))
% 189.89/24.60 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), forward_box(X, c(Y)))
% 189.89/24.60 = { by lemma 122 R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), c(forward_diamond(X, Y)))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), antidomain(forward_diamond(X, Y)))
% 189.89/24.60 = { by lemma 104 R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), c(multiplication(X, domain(Y))))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 addition(c(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, domain(Y))))
% 189.89/24.60 = { by axiom 3 (domain4) }
% 189.89/24.60 addition(antidomain(multiplication(X, Y)), antidomain(multiplication(X, antidomain(antidomain(Y)))))
% 189.89/24.60 = { by axiom 30 (domain2) }
% 189.89/24.60 antidomain(multiplication(X, antidomain(antidomain(Y))))
% 189.89/24.60 = { by axiom 3 (domain4) R->L }
% 189.89/24.60 antidomain(multiplication(X, domain(Y)))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 c(multiplication(X, domain(Y)))
% 189.89/24.60 = { by lemma 104 }
% 189.89/24.60 antidomain(forward_diamond(X, Y))
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 c(forward_diamond(X, Y))
% 189.89/24.60 = { by lemma 122 }
% 189.89/24.60 forward_box(X, c(Y))
% 189.89/24.60
% 189.89/24.60 Lemma 124: domain(domain_difference(X, c(Y))) = forward_diamond(domain(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain(domain_difference(X, c(Y)))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 domain(domain_difference(X, antidomain(Y)))
% 189.89/24.60 = { by lemma 103 }
% 189.89/24.60 forward_diamond(domain(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 125: domain(multiplication(domain(X), Y)) = forward_diamond(domain(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain(multiplication(domain(X), Y))
% 189.89/24.60 = { by lemma 87 R->L }
% 189.89/24.60 c(c(multiplication(domain(X), Y)))
% 189.89/24.60 = { by lemma 118 R->L }
% 189.89/24.60 c(c(multiplication(domain_difference(X, c(Y)), Y)))
% 189.89/24.60 = { by lemma 121 R->L }
% 189.89/24.60 c(addition(c(domain_difference(X, c(Y))), c(multiplication(domain_difference(X, c(Y)), Y))))
% 189.89/24.60 = { by lemma 118 }
% 189.89/24.60 c(addition(c(domain_difference(X, c(Y))), c(multiplication(domain(X), Y))))
% 189.89/24.60 = { by lemma 109 }
% 189.89/24.60 c(addition(forward_box(domain(X), c(Y)), c(multiplication(domain(X), Y))))
% 189.89/24.60 = { by lemma 123 }
% 189.89/24.60 c(forward_box(domain(X), c(Y)))
% 189.89/24.60 = { by lemma 106 }
% 189.89/24.60 forward_diamond(domain(X), c(c(Y)))
% 189.89/24.60 = { by lemma 101 }
% 189.89/24.60 domain(domain_difference(X, c(Y)))
% 189.89/24.60 = { by lemma 124 }
% 189.89/24.60 forward_diamond(domain(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 126: domain(multiplication(c(X), Y)) = forward_diamond(c(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain(multiplication(c(X), Y))
% 189.89/24.60 = { by lemma 68 R->L }
% 189.89/24.60 domain(multiplication(domain(c(X)), Y))
% 189.89/24.60 = { by lemma 125 }
% 189.89/24.60 forward_diamond(domain(c(X)), Y)
% 189.89/24.60 = { by lemma 68 }
% 189.89/24.60 forward_diamond(c(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 127: multiplication(domain(X), c(Y)) = domain_difference(X, Y).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(domain(X), c(Y))
% 189.89/24.60 = { by axiom 2 (complement) }
% 189.89/24.60 multiplication(domain(X), antidomain(domain(Y)))
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 domain_difference(X, domain(Y))
% 189.89/24.60 = { by lemma 89 }
% 189.89/24.60 domain_difference(X, Y)
% 189.89/24.60
% 189.89/24.60 Lemma 128: domain_difference(multiplication(X, Y), X) = zero.
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(multiplication(X, Y), X)
% 189.89/24.60 = { by lemma 127 R->L }
% 189.89/24.60 multiplication(domain(multiplication(X, Y)), c(X))
% 189.89/24.60 = { by lemma 120 R->L }
% 189.89/24.60 multiplication(domain(multiplication(X, Y)), domain_difference(c(X), multiplication(X, Y)))
% 189.89/24.60 = { by lemma 108 }
% 189.89/24.60 zero
% 189.89/24.60
% 189.89/24.60 Lemma 129: multiplication(domain(X), multiplication(c(Y), Z)) = multiplication(domain_difference(X, Y), Z).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(domain(X), multiplication(c(Y), Z))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 multiplication(domain(X), multiplication(antidomain(Y), Z))
% 189.89/24.60 = { by axiom 21 (multiplicative_associativity) }
% 189.89/24.60 multiplication(multiplication(domain(X), antidomain(Y)), Z)
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 multiplication(domain_difference(X, Y), Z)
% 189.89/24.60
% 189.89/24.60 Lemma 130: forward_diamond(c(X), c(Y)) = domain_difference(c(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 forward_diamond(c(X), c(Y))
% 189.89/24.60 = { by lemma 102 R->L }
% 189.89/24.60 domain(domain_difference(c(X), Y))
% 189.89/24.60 = { by lemma 111 R->L }
% 189.89/24.60 domain_difference(domain_difference(c(X), Y), Y)
% 189.89/24.60 = { by lemma 90 R->L }
% 189.89/24.60 domain_difference(multiplication(c(X), c(Y)), Y)
% 189.89/24.60 = { by lemma 115 }
% 189.89/24.60 domain_difference(forward_diamond(c(X), c(Y)), Y)
% 189.89/24.60 = { by lemma 116 R->L }
% 189.89/24.60 multiplication(forward_diamond(c(X), c(Y)), c(Y))
% 189.89/24.60 = { by lemma 126 R->L }
% 189.89/24.60 multiplication(domain(multiplication(c(X), c(Y))), c(Y))
% 189.89/24.60 = { by lemma 76 R->L }
% 189.89/24.60 multiplication(addition(domain_difference(multiplication(c(X), c(Y)), X), domain_difference(multiplication(c(X), c(Y)), c(X))), c(Y))
% 189.89/24.60 = { by lemma 128 }
% 189.89/24.60 multiplication(addition(domain_difference(multiplication(c(X), c(Y)), X), zero), c(Y))
% 189.89/24.60 = { by axiom 6 (additive_identity) }
% 189.89/24.60 multiplication(domain_difference(multiplication(c(X), c(Y)), X), c(Y))
% 189.89/24.60 = { by lemma 129 R->L }
% 189.89/24.60 multiplication(domain(multiplication(c(X), c(Y))), multiplication(c(X), c(Y)))
% 189.89/24.60 = { by lemma 34 }
% 189.89/24.60 multiplication(c(X), c(Y))
% 189.89/24.60 = { by lemma 90 }
% 189.89/24.60 domain_difference(c(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 131: multiplication(c(X), antidomain(Y)) = domain_difference(antidomain(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(c(X), antidomain(Y))
% 189.89/24.60 = { by lemma 31 R->L }
% 189.89/24.60 multiplication(domain(antidomain(X)), antidomain(Y))
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 domain_difference(antidomain(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 132: domain_difference(domain(X), Y) = domain_difference(X, Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain_difference(domain(X), Y)
% 189.89/24.60 = { by lemma 81 R->L }
% 189.89/24.60 domain_difference(forward_diamond(one, X), Y)
% 189.89/24.60 = { by lemma 77 R->L }
% 189.89/24.60 domain_difference(c(antidomain(X)), Y)
% 189.89/24.60 = { by lemma 48 R->L }
% 189.89/24.60 domain_difference(antidomain(c(X)), Y)
% 189.89/24.60 = { by lemma 131 R->L }
% 189.89/24.60 multiplication(c(c(X)), antidomain(Y))
% 189.89/24.60 = { by lemma 87 }
% 189.89/24.60 multiplication(domain(X), antidomain(Y))
% 189.89/24.60 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.60 domain_difference(X, Y)
% 189.89/24.60
% 189.89/24.60 Lemma 133: domain(domain_difference(X, Y)) = domain_difference(X, Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain(domain_difference(X, Y))
% 189.89/24.60 = { by lemma 101 R->L }
% 189.89/24.60 forward_diamond(domain(X), c(Y))
% 189.89/24.60 = { by lemma 87 R->L }
% 189.89/24.60 forward_diamond(c(c(X)), c(Y))
% 189.89/24.60 = { by lemma 130 }
% 189.89/24.60 domain_difference(c(c(X)), Y)
% 189.89/24.60 = { by lemma 87 }
% 189.89/24.60 domain_difference(domain(X), Y)
% 189.89/24.60 = { by lemma 132 }
% 189.89/24.60 domain_difference(X, Y)
% 189.89/24.60
% 189.89/24.60 Lemma 134: domain(backward_box(X, Y)) = backward_box(X, Y).
% 189.89/24.60 Proof:
% 189.89/24.60 domain(backward_box(X, Y))
% 189.89/24.60 = { by axiom 17 (backward_box) }
% 189.89/24.60 domain(c(backward_diamond(X, c(Y))))
% 189.89/24.60 = { by lemma 68 }
% 189.89/24.60 c(backward_diamond(X, c(Y)))
% 189.89/24.60 = { by axiom 17 (backward_box) R->L }
% 189.89/24.60 backward_box(X, Y)
% 189.89/24.60
% 189.89/24.60 Lemma 135: multiplication(c(X), addition(Y, X)) = multiplication(c(X), Y).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(c(X), addition(Y, X))
% 189.89/24.60 = { by lemma 37 R->L }
% 189.89/24.60 multiplication(antidomain(X), addition(Y, X))
% 189.89/24.60 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.60 multiplication(antidomain(X), addition(X, Y))
% 189.89/24.60 = { by lemma 36 }
% 189.89/24.60 multiplication(antidomain(X), Y)
% 189.89/24.60 = { by lemma 37 }
% 189.89/24.60 multiplication(c(X), Y)
% 189.89/24.60
% 189.89/24.60 Lemma 136: c(addition(X, one)) = zero.
% 189.89/24.60 Proof:
% 189.89/24.60 c(addition(X, one))
% 189.89/24.60 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.60 c(addition(one, X))
% 189.89/24.60 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.60 multiplication(c(addition(one, X)), one)
% 189.89/24.60 = { by lemma 135 R->L }
% 189.89/24.60 multiplication(c(addition(one, X)), addition(one, addition(one, X)))
% 189.89/24.60 = { by lemma 59 }
% 189.89/24.60 multiplication(c(addition(one, X)), addition(one, X))
% 189.89/24.60 = { by axiom 2 (complement) }
% 189.89/24.60 multiplication(antidomain(domain(addition(one, X))), addition(one, X))
% 189.89/24.60 = { by lemma 34 R->L }
% 189.89/24.60 multiplication(antidomain(domain(addition(one, X))), multiplication(domain(addition(one, X)), addition(one, X)))
% 189.89/24.60 = { by lemma 119 }
% 189.89/24.60 zero
% 189.89/24.60
% 189.89/24.60 Lemma 137: multiplication(X, addition(Y, codomain(X))) = addition(X, multiplication(X, Y)).
% 189.89/24.60 Proof:
% 189.89/24.60 multiplication(X, addition(Y, codomain(X)))
% 189.89/24.60 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.60 multiplication(X, addition(codomain(X), Y))
% 189.89/24.60 = { by axiom 25 (right_distributivity) }
% 189.89/24.60 addition(multiplication(X, codomain(X)), multiplication(X, Y))
% 189.89/24.60 = { by lemma 53 }
% 189.89/24.60 addition(X, multiplication(X, Y))
% 189.89/24.61
% 189.89/24.61 Lemma 138: codomain(forward_diamond(X, Y)) = forward_diamond(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 codomain(forward_diamond(X, Y))
% 189.89/24.61 = { by axiom 18 (forward_diamond) }
% 189.89/24.61 codomain(domain(multiplication(X, domain(Y))))
% 189.89/24.61 = { by lemma 94 }
% 189.89/24.61 domain(multiplication(X, domain(Y)))
% 189.89/24.61 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.61 forward_diamond(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 139: coantidomain(forward_diamond(X, Y)) = forward_box(X, c(Y)).
% 189.89/24.61 Proof:
% 189.89/24.61 coantidomain(forward_diamond(X, Y))
% 189.89/24.61 = { by lemma 96 R->L }
% 189.89/24.61 c(codomain(forward_diamond(X, Y)))
% 189.89/24.61 = { by lemma 99 R->L }
% 189.89/24.61 forward_box(codomain(forward_diamond(X, Y)), zero)
% 189.89/24.61 = { by lemma 136 R->L }
% 189.89/24.61 forward_box(codomain(forward_diamond(X, Y)), c(addition(forward_diamond(X, Y), one)))
% 189.89/24.61 = { by lemma 123 R->L }
% 189.89/24.61 addition(forward_box(codomain(forward_diamond(X, Y)), c(addition(forward_diamond(X, Y), one))), c(multiplication(codomain(forward_diamond(X, Y)), addition(forward_diamond(X, Y), one))))
% 189.89/24.61 = { by lemma 82 }
% 189.89/24.61 addition(forward_box(codomain(forward_diamond(X, Y)), c(addition(forward_diamond(X, Y), one))), c(addition(codomain(forward_diamond(X, Y)), multiplication(codomain(forward_diamond(X, Y)), forward_diamond(X, Y)))))
% 189.89/24.61 = { by lemma 136 }
% 189.89/24.61 addition(forward_box(codomain(forward_diamond(X, Y)), zero), c(addition(codomain(forward_diamond(X, Y)), multiplication(codomain(forward_diamond(X, Y)), forward_diamond(X, Y)))))
% 189.89/24.61 = { by lemma 99 }
% 189.89/24.61 addition(c(codomain(forward_diamond(X, Y))), c(addition(codomain(forward_diamond(X, Y)), multiplication(codomain(forward_diamond(X, Y)), forward_diamond(X, Y)))))
% 189.89/24.61 = { by lemma 137 R->L }
% 189.89/24.61 addition(c(codomain(forward_diamond(X, Y))), c(multiplication(codomain(forward_diamond(X, Y)), addition(forward_diamond(X, Y), codomain(codomain(forward_diamond(X, Y)))))))
% 189.89/24.61 = { by lemma 121 }
% 189.89/24.61 c(multiplication(codomain(forward_diamond(X, Y)), addition(forward_diamond(X, Y), codomain(codomain(forward_diamond(X, Y))))))
% 189.89/24.61 = { by lemma 137 }
% 189.89/24.61 c(addition(codomain(forward_diamond(X, Y)), multiplication(codomain(forward_diamond(X, Y)), forward_diamond(X, Y))))
% 189.89/24.61 = { by lemma 113 R->L }
% 189.89/24.61 c(addition(codomain(forward_diamond(X, Y)), multiplication(codomain(domain(forward_diamond(X, Y))), forward_diamond(X, Y))))
% 189.89/24.61 = { by lemma 84 }
% 189.89/24.61 c(addition(codomain(forward_diamond(X, Y)), forward_diamond(X, Y)))
% 189.89/24.61 = { by lemma 138 }
% 189.89/24.61 c(addition(forward_diamond(X, Y), forward_diamond(X, Y)))
% 189.89/24.61 = { by axiom 4 (additive_idempotence) }
% 189.89/24.61 c(forward_diamond(X, Y))
% 189.89/24.61 = { by lemma 122 }
% 189.89/24.61 forward_box(X, c(Y))
% 189.89/24.61
% 189.89/24.61 Lemma 140: coantidomain(multiplication(codomain(X), Y)) = coantidomain(backward_diamond(Y, X)).
% 189.89/24.61 Proof:
% 189.89/24.61 coantidomain(multiplication(codomain(X), Y))
% 189.89/24.61 = { by lemma 54 R->L }
% 189.89/24.61 codomain(coantidomain(multiplication(codomain(X), Y)))
% 189.89/24.61 = { by lemma 50 R->L }
% 189.89/24.61 coantidomain(codomain(multiplication(codomain(X), Y)))
% 189.89/24.61 = { by axiom 15 (backward_diamond) R->L }
% 189.89/24.61 coantidomain(backward_diamond(Y, X))
% 189.89/24.61
% 189.89/24.61 Lemma 141: forward_box(X, coantidomain(X)) = c(X).
% 189.89/24.61 Proof:
% 189.89/24.61 forward_box(X, coantidomain(X))
% 189.89/24.61 = { by lemma 105 R->L }
% 189.89/24.61 c(multiplication(X, c(coantidomain(X))))
% 189.89/24.61 = { by lemma 58 }
% 189.89/24.61 c(X)
% 189.89/24.61
% 189.89/24.61 Lemma 142: forward_box(codomain(X), coantidomain(X)) = coantidomain(X).
% 189.89/24.61 Proof:
% 189.89/24.61 forward_box(codomain(X), coantidomain(X))
% 189.89/24.61 = { by lemma 55 R->L }
% 189.89/24.61 forward_box(codomain(X), coantidomain(codomain(X)))
% 189.89/24.61 = { by lemma 141 }
% 189.89/24.61 c(codomain(X))
% 189.89/24.61 = { by lemma 64 }
% 189.89/24.61 backward_box(X, zero)
% 189.89/24.61 = { by lemma 70 }
% 189.89/24.61 coantidomain(X)
% 189.89/24.61
% 189.89/24.61 Lemma 143: backward_diamond(X, codomain(Y)) = backward_diamond(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 backward_diamond(X, codomain(Y))
% 189.89/24.61 = { by axiom 1 (codomain4) }
% 189.89/24.61 backward_diamond(X, coantidomain(coantidomain(Y)))
% 189.89/24.61 = { by axiom 15 (backward_diamond) }
% 189.89/24.61 codomain(multiplication(codomain(coantidomain(coantidomain(Y))), X))
% 189.89/24.61 = { by lemma 54 }
% 189.89/24.61 codomain(multiplication(coantidomain(coantidomain(Y)), X))
% 189.89/24.61 = { by axiom 1 (codomain4) R->L }
% 189.89/24.61 codomain(multiplication(codomain(Y), X))
% 189.89/24.61 = { by axiom 15 (backward_diamond) R->L }
% 189.89/24.61 backward_diamond(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 144: c(backward_diamond(X, Y)) = backward_box(X, coantidomain(Y)).
% 189.89/24.61 Proof:
% 189.89/24.61 c(backward_diamond(X, Y))
% 189.89/24.61 = { by lemma 143 R->L }
% 189.89/24.61 c(backward_diamond(X, codomain(Y)))
% 189.89/24.61 = { by lemma 67 R->L }
% 189.89/24.61 c(backward_diamond(X, c(coantidomain(Y))))
% 189.89/24.61 = { by axiom 17 (backward_box) R->L }
% 189.89/24.61 backward_box(X, coantidomain(Y))
% 189.89/24.61
% 189.89/24.61 Lemma 145: coantidomain(backward_diamond(X, Y)) = backward_box(X, coantidomain(Y)).
% 189.89/24.61 Proof:
% 189.89/24.61 coantidomain(backward_diamond(X, Y))
% 189.89/24.61 = { by lemma 140 R->L }
% 189.89/24.61 coantidomain(multiplication(codomain(Y), X))
% 189.89/24.61 = { by lemma 142 R->L }
% 189.89/24.61 forward_box(codomain(multiplication(codomain(Y), X)), coantidomain(multiplication(codomain(Y), X)))
% 189.89/24.61 = { by axiom 15 (backward_diamond) R->L }
% 189.89/24.61 forward_box(backward_diamond(X, Y), coantidomain(multiplication(codomain(Y), X)))
% 189.89/24.61 = { by lemma 140 }
% 189.89/24.61 forward_box(backward_diamond(X, Y), coantidomain(backward_diamond(X, Y)))
% 189.89/24.61 = { by lemma 141 }
% 189.89/24.61 c(backward_diamond(X, Y))
% 189.89/24.61 = { by lemma 144 }
% 189.89/24.61 backward_box(X, coantidomain(Y))
% 189.89/24.61
% 189.89/24.61 Lemma 146: backward_diamond(X, domain(X)) = codomain(X).
% 189.89/24.61 Proof:
% 189.89/24.61 backward_diamond(X, domain(X))
% 189.89/24.61 = { by axiom 15 (backward_diamond) }
% 189.89/24.61 codomain(multiplication(codomain(domain(X)), X))
% 189.89/24.61 = { by lemma 84 }
% 189.89/24.61 codomain(X)
% 189.89/24.61
% 189.89/24.61 Lemma 147: codomain(domain_difference(X, Y)) = domain_difference(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 codomain(domain_difference(X, Y))
% 189.89/24.61 = { by lemma 146 R->L }
% 189.89/24.61 backward_diamond(domain_difference(X, Y), domain(domain_difference(X, Y)))
% 189.89/24.61 = { by lemma 133 R->L }
% 189.89/24.61 backward_diamond(domain(domain_difference(X, Y)), domain(domain_difference(X, Y)))
% 189.89/24.61 = { by lemma 81 R->L }
% 189.89/24.61 backward_diamond(domain(domain_difference(X, Y)), forward_diamond(one, domain_difference(X, Y)))
% 189.89/24.61 = { by lemma 49 R->L }
% 189.89/24.61 backward_diamond(domain(domain_difference(X, Y)), domain(domain(domain_difference(X, Y))))
% 189.89/24.61 = { by lemma 146 }
% 189.89/24.61 codomain(domain(domain_difference(X, Y)))
% 189.89/24.61 = { by lemma 94 }
% 189.89/24.61 domain(domain_difference(X, Y))
% 189.89/24.61 = { by lemma 133 }
% 189.89/24.61 domain_difference(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 148: forward_box(c(X), X) = domain(X).
% 189.89/24.61 Proof:
% 189.89/24.61 forward_box(c(X), X)
% 189.89/24.61 = { by lemma 68 R->L }
% 189.89/24.61 forward_box(domain(c(X)), X)
% 189.89/24.61 = { by lemma 105 R->L }
% 189.89/24.61 c(multiplication(domain(c(X)), c(X)))
% 189.89/24.61 = { by lemma 34 }
% 189.89/24.61 c(c(X))
% 189.89/24.61 = { by lemma 87 }
% 189.89/24.61 domain(X)
% 189.89/24.61
% 189.89/24.61 Lemma 149: forward_box(X, domain(Y)) = forward_box(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 forward_box(X, domain(Y))
% 189.89/24.61 = { by axiom 16 (forward_box) }
% 189.89/24.61 c(forward_diamond(X, c(domain(Y))))
% 189.89/24.61 = { by lemma 95 }
% 189.89/24.61 c(forward_diamond(X, c(Y)))
% 189.89/24.61 = { by axiom 16 (forward_box) R->L }
% 189.89/24.61 forward_box(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 150: backward_box(X, domain(Y)) = backward_box(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 backward_box(X, domain(Y))
% 189.89/24.61 = { by axiom 17 (backward_box) }
% 189.89/24.61 c(backward_diamond(X, c(domain(Y))))
% 189.89/24.61 = { by lemma 95 }
% 189.89/24.61 c(backward_diamond(X, c(Y)))
% 189.89/24.61 = { by axiom 17 (backward_box) R->L }
% 189.89/24.61 backward_box(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 151: backward_box(X, backward_box(one, Y)) = backward_box(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 backward_box(X, backward_box(one, Y))
% 189.89/24.61 = { by lemma 74 R->L }
% 189.89/24.61 backward_box(X, coantidomain(c(Y)))
% 189.89/24.61 = { by lemma 95 R->L }
% 189.89/24.61 backward_box(X, coantidomain(c(domain(Y))))
% 189.89/24.61 = { by lemma 148 R->L }
% 189.89/24.61 backward_box(X, coantidomain(c(forward_box(c(Y), Y))))
% 189.89/24.61 = { by lemma 106 }
% 189.89/24.61 backward_box(X, coantidomain(forward_diamond(c(Y), c(Y))))
% 189.89/24.61 = { by lemma 144 R->L }
% 189.89/24.61 c(backward_diamond(X, forward_diamond(c(Y), c(Y))))
% 189.89/24.61 = { by axiom 18 (forward_diamond) }
% 189.89/24.61 c(backward_diamond(X, domain(multiplication(c(Y), domain(c(Y))))))
% 189.89/24.61 = { by lemma 87 R->L }
% 189.89/24.61 c(backward_diamond(X, c(c(multiplication(c(Y), domain(c(Y)))))))
% 189.89/24.61 = { by axiom 17 (backward_box) R->L }
% 189.89/24.61 backward_box(X, c(multiplication(c(Y), domain(c(Y)))))
% 189.89/24.61 = { by axiom 2 (complement) }
% 189.89/24.61 backward_box(X, antidomain(domain(multiplication(c(Y), domain(c(Y))))))
% 189.89/24.61 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.61 backward_box(X, antidomain(forward_diamond(c(Y), c(Y))))
% 189.89/24.61 = { by lemma 37 }
% 189.89/24.61 backward_box(X, c(forward_diamond(c(Y), c(Y))))
% 189.89/24.61 = { by lemma 122 }
% 189.89/24.61 backward_box(X, forward_box(c(Y), c(c(Y))))
% 189.89/24.61 = { by lemma 87 }
% 189.89/24.61 backward_box(X, forward_box(c(Y), domain(Y)))
% 189.89/24.61 = { by lemma 149 }
% 189.89/24.61 backward_box(X, forward_box(c(Y), Y))
% 189.89/24.61 = { by lemma 148 }
% 189.89/24.61 backward_box(X, domain(Y))
% 189.89/24.61 = { by lemma 150 }
% 189.89/24.61 backward_box(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 152: backward_box(domain(X), zero) = backward_box(one, c(X)).
% 189.89/24.61 Proof:
% 189.89/24.61 backward_box(domain(X), zero)
% 189.89/24.61 = { by lemma 81 R->L }
% 189.89/24.61 backward_box(forward_diamond(one, X), zero)
% 189.89/24.61 = { by lemma 77 R->L }
% 189.89/24.61 backward_box(c(antidomain(X)), zero)
% 189.89/24.61 = { by lemma 73 }
% 189.89/24.61 backward_box(one, antidomain(X))
% 189.89/24.61 = { by lemma 37 }
% 189.89/24.61 backward_box(one, c(X))
% 189.89/24.61
% 189.89/24.61 Lemma 153: forward_diamond(X, domain(Y)) = forward_diamond(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 forward_diamond(X, domain(Y))
% 189.89/24.61 = { by lemma 81 R->L }
% 189.89/24.61 forward_diamond(X, forward_diamond(one, Y))
% 189.89/24.61 = { by lemma 77 R->L }
% 189.89/24.61 forward_diamond(X, c(antidomain(Y)))
% 189.89/24.61 = { by lemma 48 R->L }
% 189.89/24.61 forward_diamond(X, antidomain(c(Y)))
% 189.89/24.61 = { by lemma 100 R->L }
% 189.89/24.61 domain(multiplication(X, c(c(Y))))
% 189.89/24.61 = { by lemma 87 }
% 189.89/24.61 domain(multiplication(X, domain(Y)))
% 189.89/24.61 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.61 forward_diamond(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 154: c(backward_box(X, Y)) = backward_diamond(X, c(Y)).
% 189.89/24.61 Proof:
% 189.89/24.61 c(backward_box(X, Y))
% 189.89/24.61 = { by lemma 151 R->L }
% 189.89/24.61 c(backward_box(X, backward_box(one, Y)))
% 189.89/24.61 = { by lemma 150 R->L }
% 189.89/24.61 c(backward_box(X, backward_box(one, domain(Y))))
% 189.89/24.61 = { by lemma 81 R->L }
% 189.89/24.61 c(backward_box(X, backward_box(one, forward_diamond(one, Y))))
% 189.89/24.61 = { by lemma 77 R->L }
% 189.89/24.61 c(backward_box(X, backward_box(one, c(antidomain(Y)))))
% 189.89/24.61 = { by lemma 134 R->L }
% 189.89/24.61 c(backward_box(X, domain(backward_box(one, c(antidomain(Y))))))
% 189.89/24.61 = { by lemma 152 R->L }
% 189.89/24.61 c(backward_box(X, domain(backward_box(domain(antidomain(Y)), zero))))
% 189.89/24.61 = { by lemma 64 R->L }
% 189.89/24.61 c(backward_box(X, domain(c(codomain(domain(antidomain(Y)))))))
% 189.89/24.61 = { by axiom 7 (multiplicative_right_identity) R->L }
% 189.89/24.61 c(backward_box(X, domain(multiplication(c(codomain(domain(antidomain(Y)))), one))))
% 189.89/24.61 = { by lemma 83 R->L }
% 189.89/24.61 c(backward_box(X, domain(multiplication(c(codomain(domain(antidomain(Y)))), addition(c(antidomain(Y)), codomain(domain(antidomain(Y))))))))
% 189.89/24.61 = { by lemma 135 }
% 189.89/24.61 c(backward_box(X, domain(multiplication(c(codomain(domain(antidomain(Y)))), c(antidomain(Y))))))
% 189.89/24.61 = { by lemma 31 R->L }
% 189.89/24.61 c(backward_box(X, domain(multiplication(domain(antidomain(codomain(domain(antidomain(Y))))), c(antidomain(Y))))))
% 189.89/24.61 = { by lemma 88 }
% 189.89/24.61 c(backward_box(X, domain(domain_difference(antidomain(codomain(domain(antidomain(Y)))), domain(antidomain(Y))))))
% 189.89/24.61 = { by lemma 89 }
% 189.89/24.61 c(backward_box(X, domain(domain_difference(antidomain(codomain(domain(antidomain(Y)))), antidomain(Y)))))
% 189.89/24.61 = { by lemma 37 }
% 189.89/24.61 c(backward_box(X, domain(domain_difference(c(codomain(domain(antidomain(Y)))), antidomain(Y)))))
% 189.89/24.61 = { by lemma 64 }
% 189.89/24.61 c(backward_box(X, domain(domain_difference(backward_box(domain(antidomain(Y)), zero), antidomain(Y)))))
% 189.89/24.61 = { by lemma 152 }
% 189.89/24.61 c(backward_box(X, domain(domain_difference(backward_box(one, c(antidomain(Y))), antidomain(Y)))))
% 189.89/24.61 = { by lemma 103 }
% 189.89/24.61 c(backward_box(X, forward_diamond(domain(backward_box(one, c(antidomain(Y)))), Y)))
% 189.89/24.61 = { by lemma 134 }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, c(antidomain(Y))), Y)))
% 189.89/24.61 = { by lemma 77 }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, forward_diamond(one, Y)), Y)))
% 189.89/24.61 = { by lemma 81 }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, domain(Y)), Y)))
% 189.89/24.61 = { by lemma 150 }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, Y), Y)))
% 189.89/24.61 = { by lemma 153 R->L }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, Y), domain(Y))))
% 189.89/24.61 = { by lemma 81 R->L }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, Y), forward_diamond(one, Y))))
% 189.89/24.61 = { by lemma 77 R->L }
% 189.89/24.61 c(backward_box(X, forward_diamond(backward_box(one, Y), c(antidomain(Y)))))
% 189.89/24.61 = { by lemma 37 R->L }
% 189.89/24.61 antidomain(backward_box(X, forward_diamond(backward_box(one, Y), c(antidomain(Y)))))
% 189.89/24.61 = { by axiom 17 (backward_box) }
% 189.89/24.61 antidomain(c(backward_diamond(X, c(forward_diamond(backward_box(one, Y), c(antidomain(Y)))))))
% 189.89/24.61 = { by axiom 16 (forward_box) R->L }
% 189.89/24.61 antidomain(c(backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y)))))
% 189.89/24.61 = { by lemma 47 }
% 189.89/24.61 domain(domain(backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y)))))
% 189.89/24.61 = { by lemma 49 }
% 189.89/24.61 forward_diamond(one, backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y))))
% 189.89/24.61 = { by lemma 81 }
% 189.89/24.61 domain(backward_diamond(X, forward_box(backward_box(one, Y), antidomain(Y))))
% 189.89/24.61 = { by lemma 37 }
% 189.89/24.61 domain(backward_diamond(X, forward_box(backward_box(one, Y), c(Y))))
% 189.89/24.61 = { by lemma 74 R->L }
% 189.89/24.61 domain(backward_diamond(X, forward_box(coantidomain(c(Y)), c(Y))))
% 189.89/24.61 = { by lemma 68 R->L }
% 189.89/24.61 domain(backward_diamond(X, forward_box(coantidomain(domain(c(Y))), c(Y))))
% 189.89/24.61 = { by lemma 105 R->L }
% 189.89/24.61 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), c(c(Y))))))
% 189.89/24.61 = { by lemma 46 R->L }
% 189.89/24.61 domain(backward_diamond(X, c(addition(zero, multiplication(coantidomain(domain(c(Y))), c(c(Y)))))))
% 189.89/24.61 = { by axiom 11 (codomain1) R->L }
% 189.89/24.61 domain(backward_diamond(X, c(addition(multiplication(coantidomain(domain(c(Y))), coantidomain(coantidomain(domain(c(Y))))), multiplication(coantidomain(domain(c(Y))), c(c(Y)))))))
% 189.89/24.61 = { by axiom 1 (codomain4) R->L }
% 189.89/24.61 domain(backward_diamond(X, c(addition(multiplication(coantidomain(domain(c(Y))), codomain(domain(c(Y)))), multiplication(coantidomain(domain(c(Y))), c(c(Y)))))))
% 189.89/24.61 = { by axiom 25 (right_distributivity) R->L }
% 189.89/24.61 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), addition(codomain(domain(c(Y))), c(c(Y)))))))
% 189.89/24.61 = { by axiom 5 (additive_commutativity) }
% 189.89/24.61 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), addition(c(c(Y)), codomain(domain(c(Y))))))))
% 189.89/24.61 = { by lemma 83 }
% 189.89/24.61 domain(backward_diamond(X, c(multiplication(coantidomain(domain(c(Y))), one))))
% 189.89/24.61 = { by axiom 7 (multiplicative_right_identity) }
% 189.89/24.61 domain(backward_diamond(X, c(coantidomain(domain(c(Y))))))
% 189.89/24.61 = { by lemma 67 }
% 189.89/24.61 domain(backward_diamond(X, codomain(domain(c(Y)))))
% 189.89/24.61 = { by lemma 68 }
% 189.89/24.61 domain(backward_diamond(X, codomain(c(Y))))
% 189.89/24.61 = { by axiom 15 (backward_diamond) }
% 189.89/24.61 domain(codomain(multiplication(codomain(codomain(c(Y))), X)))
% 189.89/24.61 = { by lemma 69 }
% 189.89/24.61 codomain(multiplication(codomain(codomain(c(Y))), X))
% 189.89/24.61 = { by axiom 15 (backward_diamond) R->L }
% 189.89/24.61 backward_diamond(X, codomain(c(Y)))
% 189.89/24.61 = { by lemma 143 }
% 189.89/24.61 backward_diamond(X, c(Y))
% 189.89/24.61
% 189.89/24.61 Lemma 155: domain_difference(forward_diamond(X, Y), c(X)) = forward_diamond(X, Y).
% 189.89/24.61 Proof:
% 189.89/24.61 domain_difference(forward_diamond(X, Y), c(X))
% 189.89/24.61 = { by lemma 46 R->L }
% 189.89/24.61 addition(zero, domain_difference(forward_diamond(X, Y), c(X)))
% 189.89/24.61 = { by lemma 128 R->L }
% 189.89/24.61 addition(domain_difference(multiplication(X, domain(Y)), X), domain_difference(forward_diamond(X, Y), c(X)))
% 189.89/24.61 = { by lemma 114 }
% 189.89/24.61 addition(domain_difference(forward_diamond(X, Y), X), domain_difference(forward_diamond(X, Y), c(X)))
% 189.89/24.61 = { by lemma 76 }
% 189.89/24.61 domain(forward_diamond(X, Y))
% 189.89/24.61 = { by lemma 113 }
% 189.89/24.61 forward_diamond(X, Y)
% 189.89/24.61
% 189.89/24.61 Lemma 156: multiplication(c(X), domain(Y)) = domain_difference(c(X), c(Y)).
% 189.89/24.61 Proof:
% 189.89/24.61 multiplication(c(X), domain(Y))
% 189.89/24.61 = { by lemma 31 R->L }
% 189.89/24.62 multiplication(domain(antidomain(X)), domain(Y))
% 189.89/24.62 = { by lemma 78 }
% 189.89/24.62 domain_difference(antidomain(X), antidomain(Y))
% 189.89/24.62 = { by lemma 37 }
% 189.89/24.62 domain_difference(c(X), antidomain(Y))
% 189.89/24.62 = { by lemma 37 }
% 189.89/24.62 domain_difference(c(X), c(Y))
% 189.89/24.62
% 189.89/24.62 Lemma 157: multiplication(c(X), domain(X)) = zero.
% 189.89/24.62 Proof:
% 189.89/24.62 multiplication(c(X), domain(X))
% 189.89/24.62 = { by axiom 2 (complement) }
% 189.89/24.62 multiplication(antidomain(domain(X)), domain(X))
% 189.89/24.62 = { by axiom 12 (domain1) }
% 189.89/24.62 zero
% 189.89/24.62
% 189.89/24.62 Lemma 158: multiplication(c(X), addition(domain(X), Y)) = multiplication(c(X), Y).
% 189.89/24.62 Proof:
% 189.89/24.62 multiplication(c(X), addition(domain(X), Y))
% 189.89/24.62 = { by axiom 25 (right_distributivity) }
% 189.89/24.62 addition(multiplication(c(X), domain(X)), multiplication(c(X), Y))
% 189.89/24.62 = { by lemma 157 }
% 189.89/24.62 addition(zero, multiplication(c(X), Y))
% 189.89/24.62 = { by lemma 46 }
% 189.89/24.62 multiplication(c(X), Y)
% 189.89/24.62
% 189.89/24.62 Lemma 159: addition(c(X), addition(Y, domain_difference(Z, X))) = addition(Y, c(X)).
% 189.89/24.62 Proof:
% 189.89/24.62 addition(c(X), addition(Y, domain_difference(Z, X)))
% 189.89/24.62 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.62 addition(c(X), addition(domain_difference(Z, X), Y))
% 189.89/24.62 = { by axiom 19 (additive_associativity) }
% 189.89/24.62 addition(addition(c(X), domain_difference(Z, X)), Y)
% 189.89/24.62 = { by lemma 107 }
% 189.89/24.62 addition(c(X), Y)
% 189.89/24.62 = { by axiom 5 (additive_commutativity) }
% 189.89/24.62 addition(Y, c(X))
% 189.89/24.62
% 189.89/24.62 Lemma 160: addition(domain(X), domain_difference(Y, X)) = addition(domain(X), domain(Y)).
% 189.89/24.62 Proof:
% 189.89/24.62 addition(domain(X), domain_difference(Y, X))
% 189.89/24.62 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.62 addition(domain_difference(Y, X), domain(X))
% 189.89/24.62 = { by lemma 87 R->L }
% 189.89/24.62 addition(domain_difference(Y, X), c(c(X)))
% 189.89/24.62 = { by lemma 159 R->L }
% 189.89/24.62 addition(c(c(X)), addition(domain_difference(Y, X), domain_difference(Y, c(X))))
% 189.89/24.62 = { by lemma 76 }
% 189.89/24.62 addition(c(c(X)), domain(Y))
% 189.89/24.62 = { by lemma 87 }
% 189.89/24.62 addition(domain(X), domain(Y))
% 189.89/24.62
% 189.89/24.62 Lemma 161: forward_diamond(c(X), Y) = domain_difference(Y, X).
% 189.89/24.62 Proof:
% 189.89/24.62 forward_diamond(c(X), Y)
% 189.89/24.62 = { by lemma 155 R->L }
% 189.89/24.62 domain_difference(forward_diamond(c(X), Y), c(c(X)))
% 189.89/24.62 = { by lemma 68 R->L }
% 189.89/24.62 domain_difference(forward_diamond(domain(c(X)), Y), c(c(X)))
% 189.89/24.62 = { by lemma 114 R->L }
% 189.89/24.62 domain_difference(multiplication(domain(c(X)), domain(Y)), c(c(X)))
% 189.89/24.62 = { by lemma 112 R->L }
% 189.89/24.62 multiplication(forward_diamond(domain(c(X)), Y), c(c(c(X))))
% 189.89/24.62 = { by lemma 37 R->L }
% 189.89/24.62 multiplication(forward_diamond(domain(c(X)), Y), antidomain(c(c(X))))
% 189.89/24.62 = { by lemma 103 R->L }
% 189.89/24.62 multiplication(domain(domain_difference(c(X), antidomain(Y))), antidomain(c(c(X))))
% 189.89/24.62 = { by axiom 20 (domain_difference) R->L }
% 189.89/24.62 domain_difference(domain_difference(c(X), antidomain(Y)), c(c(X)))
% 189.89/24.62 = { by lemma 37 }
% 189.89/24.62 domain_difference(domain_difference(c(X), c(Y)), c(c(X)))
% 189.89/24.62 = { by lemma 156 R->L }
% 189.89/24.62 domain_difference(multiplication(c(X), domain(Y)), c(c(X)))
% 189.89/24.62 = { by lemma 158 R->L }
% 189.89/24.62 domain_difference(multiplication(c(X), addition(domain(X), domain(Y))), c(c(X)))
% 189.89/24.62 = { by lemma 160 R->L }
% 189.89/24.62 domain_difference(multiplication(c(X), addition(domain(X), domain_difference(Y, X))), c(c(X)))
% 189.89/24.62 = { by lemma 158 }
% 189.89/24.62 domain_difference(multiplication(c(X), domain_difference(Y, X)), c(c(X)))
% 189.89/24.62 = { by lemma 46 R->L }
% 189.89/24.62 addition(zero, domain_difference(multiplication(c(X), domain_difference(Y, X)), c(c(X))))
% 189.89/24.62 = { by lemma 128 R->L }
% 189.89/24.62 addition(domain_difference(multiplication(c(X), domain_difference(Y, X)), c(X)), domain_difference(multiplication(c(X), domain_difference(Y, X)), c(c(X))))
% 189.89/24.62 = { by lemma 76 }
% 189.89/24.62 domain(multiplication(c(X), domain_difference(Y, X)))
% 189.89/24.62 = { by lemma 126 }
% 189.89/24.62 forward_diamond(c(X), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 95 R->L }
% 189.89/24.62 forward_diamond(c(domain(X)), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 110 R->L }
% 189.89/24.62 forward_diamond(c(domain_difference(X, domain_difference(Y, X))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 109 }
% 189.89/24.62 forward_diamond(forward_box(domain(X), domain_difference(Y, X)), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 87 R->L }
% 189.89/24.62 forward_diamond(forward_box(c(c(X)), domain_difference(Y, X)), domain_difference(Y, X))
% 189.89/24.62 = { by axiom 18 (forward_diamond) }
% 189.89/24.62 domain(multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X))))
% 189.89/24.62 = { by lemma 46 R->L }
% 189.89/24.62 domain(addition(zero, multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 189.89/24.62 = { by axiom 8 (right_annihilation) R->L }
% 189.89/24.62 domain(addition(multiplication(domain(c(c(X))), zero), multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 189.89/24.62 = { by lemma 157 R->L }
% 189.89/24.62 domain(addition(multiplication(domain(c(c(X))), multiplication(c(domain_difference(Y, X)), domain(domain_difference(Y, X)))), multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 189.89/24.62 = { by lemma 129 }
% 189.89/24.62 domain(addition(multiplication(domain_difference(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X))), multiplication(forward_box(c(c(X)), domain_difference(Y, X)), domain(domain_difference(Y, X)))))
% 189.89/24.62 = { by axiom 26 (left_distributivity) R->L }
% 189.89/24.62 domain(multiplication(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), domain_difference(Y, X))), domain(domain_difference(Y, X))))
% 189.89/24.62 = { by axiom 5 (additive_commutativity) }
% 189.89/24.62 domain(multiplication(addition(forward_box(c(c(X)), domain_difference(Y, X)), domain_difference(c(c(X)), domain_difference(Y, X))), domain(domain_difference(Y, X))))
% 189.89/24.62 = { by axiom 18 (forward_diamond) R->L }
% 189.89/24.62 forward_diamond(addition(forward_box(c(c(X)), domain_difference(Y, X)), domain_difference(c(c(X)), domain_difference(Y, X))), domain_difference(Y, X))
% 189.89/24.62 = { by axiom 5 (additive_commutativity) }
% 189.89/24.62 forward_diamond(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), domain_difference(Y, X))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 149 R->L }
% 189.89/24.62 forward_diamond(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), domain(domain_difference(Y, X)))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 87 R->L }
% 189.89/24.62 forward_diamond(addition(domain_difference(c(c(X)), domain_difference(Y, X)), forward_box(c(c(X)), c(c(domain_difference(Y, X))))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 130 R->L }
% 189.89/24.62 forward_diamond(addition(forward_diamond(c(c(X)), c(domain_difference(Y, X))), forward_box(c(c(X)), c(c(domain_difference(Y, X))))), domain_difference(Y, X))
% 189.89/24.62 = { by axiom 5 (additive_commutativity) R->L }
% 189.89/24.62 forward_diamond(addition(forward_box(c(c(X)), c(c(domain_difference(Y, X)))), forward_diamond(c(c(X)), c(domain_difference(Y, X)))), domain_difference(Y, X))
% 189.89/24.62 = { by axiom 18 (forward_diamond) }
% 189.89/24.62 forward_diamond(addition(forward_box(c(c(X)), c(c(domain_difference(Y, X)))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 122 R->L }
% 189.89/24.62 forward_diamond(addition(c(forward_diamond(c(c(X)), c(domain_difference(Y, X)))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 37 R->L }
% 189.89/24.62 forward_diamond(addition(antidomain(forward_diamond(c(c(X)), c(domain_difference(Y, X)))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 104 R->L }
% 189.89/24.62 forward_diamond(addition(c(multiplication(c(c(X)), domain(c(domain_difference(Y, X))))), domain(multiplication(c(c(X)), domain(c(domain_difference(Y, X)))))), domain_difference(Y, X))
% 189.89/24.62 = { by lemma 57 }
% 189.89/24.62 forward_diamond(one, domain_difference(Y, X))
% 189.89/24.62 = { by lemma 81 }
% 189.89/24.62 domain(domain_difference(Y, X))
% 189.89/24.62 = { by lemma 133 }
% 189.89/24.62 domain_difference(Y, X)
% 189.89/24.62
% 189.89/24.62 Lemma 162: forward_diamond(domain(X), Y) = domain_difference(X, c(Y)).
% 189.89/24.62 Proof:
% 189.89/24.62 forward_diamond(domain(X), Y)
% 189.89/24.62 = { by lemma 124 R->L }
% 189.89/24.62 domain(domain_difference(X, c(Y)))
% 189.89/24.62 = { by lemma 133 }
% 189.89/24.62 domain_difference(X, c(Y))
% 189.89/24.62
% 189.89/24.62 Lemma 163: domain_difference(X, c(Y)) = domain_difference(Y, c(X)).
% 189.89/24.62 Proof:
% 189.89/24.62 domain_difference(X, c(Y))
% 189.89/24.62 = { by lemma 161 R->L }
% 189.89/24.62 forward_diamond(c(c(Y)), X)
% 189.89/24.62 = { by lemma 87 }
% 189.89/24.62 forward_diamond(domain(Y), X)
% 189.89/24.62 = { by lemma 162 }
% 189.89/24.62 domain_difference(Y, c(X))
% 189.89/24.62
% 189.89/24.62 Lemma 164: domain_difference(c(X), X) = c(X).
% 189.89/24.62 Proof:
% 189.89/24.62 domain_difference(c(X), X)
% 189.89/24.62 = { by lemma 37 R->L }
% 189.89/24.62 domain_difference(antidomain(X), X)
% 189.89/24.62 = { by lemma 80 }
% 189.89/24.62 antidomain(X)
% 189.89/24.62 = { by lemma 37 }
% 189.89/24.62 c(X)
% 189.89/24.62
% 189.89/24.62 Lemma 165: multiplication(domain(X), backward_box(Y, Z)) = domain_difference(X, backward_diamond(Y, c(Z))).
% 189.89/24.62 Proof:
% 189.89/24.62 multiplication(domain(X), backward_box(Y, Z))
% 189.89/24.62 = { by axiom 17 (backward_box) }
% 190.93/24.62 multiplication(domain(X), c(backward_diamond(Y, c(Z))))
% 190.93/24.62 = { by lemma 88 }
% 190.93/24.62 domain_difference(X, domain(backward_diamond(Y, c(Z))))
% 190.93/24.62 = { by lemma 89 }
% 190.93/24.62 domain_difference(X, backward_diamond(Y, c(Z)))
% 190.93/24.62
% 190.93/24.62 Lemma 166: multiplication(domain(X), coantidomain(Y)) = domain_difference(X, codomain(Y)).
% 190.93/24.62 Proof:
% 190.93/24.62 multiplication(domain(X), coantidomain(Y))
% 190.93/24.62 = { by lemma 55 R->L }
% 190.93/24.62 multiplication(domain(X), coantidomain(codomain(Y)))
% 190.93/24.62 = { by lemma 58 R->L }
% 190.93/24.62 multiplication(domain(X), coantidomain(multiplication(codomain(Y), c(coantidomain(codomain(Y))))))
% 190.93/24.62 = { by lemma 140 }
% 190.93/24.62 multiplication(domain(X), coantidomain(backward_diamond(c(coantidomain(codomain(Y))), Y)))
% 190.93/24.62 = { by lemma 145 }
% 190.93/24.62 multiplication(domain(X), backward_box(c(coantidomain(codomain(Y))), coantidomain(Y)))
% 190.93/24.62 = { by lemma 67 }
% 190.93/24.62 multiplication(domain(X), backward_box(codomain(codomain(Y)), coantidomain(Y)))
% 190.93/24.62 = { by axiom 7 (multiplicative_right_identity) R->L }
% 190.93/24.62 multiplication(domain(X), backward_box(codomain(multiplication(codomain(Y), one)), coantidomain(Y)))
% 190.93/24.62 = { by axiom 15 (backward_diamond) R->L }
% 190.93/24.62 multiplication(domain(X), backward_box(backward_diamond(one, Y), coantidomain(Y)))
% 190.93/24.62 = { by lemma 72 }
% 190.93/24.62 multiplication(domain(X), backward_box(codomain(Y), coantidomain(Y)))
% 190.93/24.62 = { by lemma 165 }
% 190.93/24.62 domain_difference(X, backward_diamond(codomain(Y), c(coantidomain(Y))))
% 190.93/24.62 = { by lemma 67 }
% 190.93/24.62 domain_difference(X, backward_diamond(codomain(Y), codomain(Y)))
% 190.93/24.62 = { by lemma 143 }
% 190.93/24.62 domain_difference(X, backward_diamond(codomain(Y), Y))
% 190.93/24.62 = { by axiom 15 (backward_diamond) }
% 190.93/24.62 domain_difference(X, codomain(multiplication(codomain(Y), codomain(Y))))
% 190.93/24.62 = { by axiom 1 (codomain4) }
% 190.93/24.62 domain_difference(X, codomain(multiplication(codomain(Y), coantidomain(coantidomain(Y)))))
% 190.93/24.62 = { by axiom 1 (codomain4) }
% 190.93/24.62 domain_difference(X, codomain(multiplication(coantidomain(coantidomain(Y)), coantidomain(coantidomain(Y)))))
% 190.93/24.62 = { by lemma 54 R->L }
% 190.93/24.62 domain_difference(X, codomain(multiplication(coantidomain(coantidomain(Y)), codomain(coantidomain(coantidomain(Y))))))
% 190.93/24.62 = { by lemma 53 }
% 190.93/24.62 domain_difference(X, codomain(coantidomain(coantidomain(Y))))
% 190.93/24.62 = { by axiom 1 (codomain4) R->L }
% 190.93/24.62 domain_difference(X, codomain(codomain(Y)))
% 190.93/24.62 = { by lemma 71 }
% 190.93/24.62 domain_difference(X, backward_diamond(one, Y))
% 190.93/24.62 = { by lemma 72 }
% 190.93/24.62 domain_difference(X, codomain(Y))
% 190.93/24.62
% 190.93/24.62 Lemma 167: domain_difference(c(X), c(Y)) = forward_diamond(c(X), Y).
% 190.93/24.62 Proof:
% 190.93/24.62 domain_difference(c(X), c(Y))
% 190.93/24.62 = { by lemma 130 R->L }
% 190.93/24.62 forward_diamond(c(X), c(c(Y)))
% 190.93/24.62 = { by lemma 87 }
% 190.93/24.62 forward_diamond(c(X), domain(Y))
% 190.93/24.62 = { by lemma 153 }
% 190.93/24.62 forward_diamond(c(X), Y)
% 190.93/24.62
% 190.93/24.62 Lemma 168: multiplication(domain(X), domain_difference(Y, Z)) = domain_difference(X, forward_box(domain(Y), Z)).
% 190.93/24.62 Proof:
% 190.93/24.62 multiplication(domain(X), domain_difference(Y, Z))
% 190.93/24.62 = { by lemma 133 R->L }
% 190.93/24.62 multiplication(domain(X), domain(domain_difference(Y, Z)))
% 190.93/24.62 = { by lemma 75 }
% 190.93/24.62 domain_difference(X, c(domain_difference(Y, Z)))
% 190.93/24.62 = { by lemma 109 }
% 190.93/24.62 domain_difference(X, forward_box(domain(Y), Z))
% 190.93/24.62
% 190.93/24.62 Lemma 169: multiplication(codomain(c(X)), c(X)) = c(X).
% 190.93/24.62 Proof:
% 190.93/24.62 multiplication(codomain(c(X)), c(X))
% 190.93/24.62 = { by lemma 37 R->L }
% 190.93/24.62 multiplication(codomain(c(X)), antidomain(X))
% 190.93/24.62 = { by lemma 31 R->L }
% 190.93/24.62 multiplication(codomain(domain(antidomain(X))), antidomain(X))
% 190.93/24.62 = { by lemma 84 }
% 190.93/24.62 antidomain(X)
% 190.93/24.62 = { by lemma 37 }
% 190.93/24.62 c(X)
% 190.93/24.62
% 190.93/24.62 Lemma 170: multiplication(domain_difference(X, Y), c(Z)) = domain_difference(X, forward_box(c(Y), Z)).
% 190.93/24.62 Proof:
% 190.93/24.62 multiplication(domain_difference(X, Y), c(Z))
% 190.93/24.62 = { by lemma 37 R->L }
% 190.93/24.62 multiplication(domain_difference(X, Y), antidomain(Z))
% 190.93/24.62 = { by lemma 129 R->L }
% 190.93/24.62 multiplication(domain(X), multiplication(c(Y), antidomain(Z)))
% 190.93/24.62 = { by lemma 131 }
% 190.93/24.62 multiplication(domain(X), domain_difference(antidomain(Y), Z))
% 190.93/24.62 = { by lemma 37 }
% 190.93/24.62 multiplication(domain(X), domain_difference(c(Y), Z))
% 190.93/24.62 = { by lemma 130 R->L }
% 190.93/24.62 multiplication(domain(X), forward_diamond(c(Y), c(Z)))
% 190.93/24.62 = { by axiom 18 (forward_diamond) }
% 190.93/24.62 multiplication(domain(X), domain(multiplication(c(Y), domain(c(Z)))))
% 190.93/24.62 = { by lemma 78 }
% 190.93/24.62 domain_difference(X, antidomain(multiplication(c(Y), domain(c(Z)))))
% 190.93/24.62 = { by lemma 37 }
% 190.93/24.62 domain_difference(X, c(multiplication(c(Y), domain(c(Z)))))
% 190.93/24.62 = { by lemma 104 }
% 190.93/24.62 domain_difference(X, antidomain(forward_diamond(c(Y), c(Z))))
% 190.93/24.62 = { by lemma 37 }
% 190.93/24.62 domain_difference(X, c(forward_diamond(c(Y), c(Z))))
% 190.93/24.62 = { by lemma 122 }
% 190.93/24.62 domain_difference(X, forward_box(c(Y), c(c(Z))))
% 190.93/24.62 = { by lemma 87 }
% 190.93/24.62 domain_difference(X, forward_box(c(Y), domain(Z)))
% 190.93/24.62 = { by lemma 149 }
% 190.93/24.62 domain_difference(X, forward_box(c(Y), Z))
% 190.93/24.62
% 190.93/24.62 Lemma 171: multiplication(forward_diamond(X, Y), codomain(Z)) = domain_difference(forward_diamond(X, Y), coantidomain(Z)).
% 190.93/24.62 Proof:
% 190.93/24.62 multiplication(forward_diamond(X, Y), codomain(Z))
% 190.93/24.62 = { by axiom 1 (codomain4) }
% 190.93/24.62 multiplication(forward_diamond(X, Y), coantidomain(coantidomain(Z)))
% 190.93/24.62 = { by lemma 70 R->L }
% 190.93/24.62 multiplication(forward_diamond(X, Y), backward_box(coantidomain(Z), zero))
% 190.93/24.62 = { by axiom 17 (backward_box) }
% 190.93/24.62 multiplication(forward_diamond(X, Y), c(backward_diamond(coantidomain(Z), c(zero))))
% 190.93/24.62 = { by lemma 116 }
% 190.93/24.62 domain_difference(forward_diamond(X, Y), backward_diamond(coantidomain(Z), c(zero)))
% 190.93/24.62 = { by lemma 45 }
% 190.93/24.62 domain_difference(forward_diamond(X, Y), backward_diamond(coantidomain(Z), one))
% 190.93/24.62 = { by lemma 63 }
% 190.93/24.62 domain_difference(forward_diamond(X, Y), codomain(coantidomain(Z)))
% 190.93/24.62 = { by lemma 54 }
% 190.93/24.63 domain_difference(forward_diamond(X, Y), coantidomain(Z))
% 190.93/24.63
% 190.93/24.63 Lemma 172: domain_difference(X, backward_diamond(Y, c(Z))) = forward_diamond(backward_box(Y, Z), X).
% 190.93/24.63 Proof:
% 190.93/24.63 domain_difference(X, backward_diamond(Y, c(Z)))
% 190.93/24.63 = { by lemma 154 R->L }
% 190.93/24.63 domain_difference(X, c(backward_box(Y, Z)))
% 190.93/24.63 = { by lemma 163 R->L }
% 190.93/24.63 domain_difference(backward_box(Y, Z), c(X))
% 190.93/24.63 = { by lemma 162 R->L }
% 190.93/24.63 forward_diamond(domain(backward_box(Y, Z)), X)
% 190.93/24.63 = { by lemma 134 }
% 190.93/24.63 forward_diamond(backward_box(Y, Z), X)
% 190.93/24.63
% 190.93/24.63 Lemma 173: multiplication(backward_box(X, Y), domain(Z)) = forward_diamond(backward_box(X, Y), Z).
% 190.93/24.63 Proof:
% 190.93/24.63 multiplication(backward_box(X, Y), domain(Z))
% 190.93/24.63 = { by axiom 17 (backward_box) }
% 190.93/24.63 multiplication(c(backward_diamond(X, c(Y))), domain(Z))
% 190.93/24.63 = { by lemma 156 }
% 190.93/24.63 domain_difference(c(backward_diamond(X, c(Y))), c(Z))
% 190.93/24.63 = { by axiom 17 (backward_box) R->L }
% 190.93/24.63 domain_difference(backward_box(X, Y), c(Z))
% 190.93/24.63 = { by lemma 163 }
% 190.93/24.63 domain_difference(Z, c(backward_box(X, Y)))
% 190.93/24.63 = { by lemma 154 }
% 190.93/24.63 domain_difference(Z, backward_diamond(X, c(Y)))
% 190.93/24.63 = { by lemma 172 }
% 190.93/24.63 forward_diamond(backward_box(X, Y), Z)
% 190.93/24.63
% 190.93/24.63 Lemma 174: forward_diamond(backward_box(X, Y), backward_box(X, Y)) = backward_box(X, Y).
% 190.93/24.63 Proof:
% 190.93/24.63 forward_diamond(backward_box(X, Y), backward_box(X, Y))
% 190.93/24.63 = { by axiom 18 (forward_diamond) }
% 190.93/24.63 domain(multiplication(backward_box(X, Y), domain(backward_box(X, Y))))
% 190.93/24.63 = { by lemma 32 R->L }
% 190.93/24.63 domain(multiplication(addition(backward_box(X, Y), antidomain(domain(backward_box(X, Y)))), domain(backward_box(X, Y))))
% 190.93/24.63 = { by axiom 18 (forward_diamond) R->L }
% 190.93/24.63 forward_diamond(addition(backward_box(X, Y), antidomain(domain(backward_box(X, Y)))), backward_box(X, Y))
% 190.93/24.63 = { by axiom 2 (complement) R->L }
% 190.93/24.63 forward_diamond(addition(backward_box(X, Y), c(backward_box(X, Y))), backward_box(X, Y))
% 190.93/24.63 = { by axiom 5 (additive_commutativity) R->L }
% 190.93/24.63 forward_diamond(addition(c(backward_box(X, Y)), backward_box(X, Y)), backward_box(X, Y))
% 190.93/24.63 = { by axiom 17 (backward_box) }
% 190.93/24.63 forward_diamond(addition(c(backward_box(X, Y)), c(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by lemma 37 R->L }
% 190.93/24.63 forward_diamond(addition(c(backward_box(X, Y)), antidomain(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by lemma 37 R->L }
% 190.93/24.63 forward_diamond(addition(antidomain(backward_box(X, Y)), antidomain(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by axiom 17 (backward_box) }
% 190.93/24.63 forward_diamond(addition(antidomain(c(backward_diamond(X, c(Y)))), antidomain(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by lemma 47 }
% 190.93/24.63 forward_diamond(addition(domain(domain(backward_diamond(X, c(Y)))), antidomain(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by lemma 49 }
% 190.93/24.63 forward_diamond(addition(forward_diamond(one, backward_diamond(X, c(Y))), antidomain(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by lemma 81 }
% 190.93/24.63 forward_diamond(addition(domain(backward_diamond(X, c(Y))), antidomain(backward_diamond(X, c(Y)))), backward_box(X, Y))
% 190.93/24.63 = { by lemma 33 }
% 190.93/24.63 forward_diamond(one, backward_box(X, Y))
% 190.93/24.63 = { by lemma 81 }
% 190.93/24.63 domain(backward_box(X, Y))
% 190.93/24.63 = { by lemma 134 }
% 190.93/24.63 backward_box(X, Y)
% 190.93/24.63
% 190.93/24.63 Lemma 175: multiplication(backward_box(X, Y), codomain(Z)) = domain_difference(backward_box(X, Y), coantidomain(Z)).
% 190.93/24.63 Proof:
% 190.93/24.63 multiplication(backward_box(X, Y), codomain(Z))
% 190.93/24.63 = { by lemma 174 R->L }
% 190.93/24.63 multiplication(forward_diamond(backward_box(X, Y), backward_box(X, Y)), codomain(Z))
% 190.93/24.63 = { by lemma 171 }
% 190.93/24.63 domain_difference(forward_diamond(backward_box(X, Y), backward_box(X, Y)), coantidomain(Z))
% 190.93/24.63 = { by lemma 174 }
% 190.93/24.63 domain_difference(backward_box(X, Y), coantidomain(Z))
% 190.93/24.63
% 190.93/24.63 Lemma 176: domain_difference(X, forward_box(domain(Y), Z)) = forward_diamond(domain_difference(Y, Z), X).
% 190.93/24.63 Proof:
% 190.93/24.63 domain_difference(X, forward_box(domain(Y), Z))
% 190.93/24.63 = { by lemma 109 R->L }
% 190.93/24.63 domain_difference(X, c(domain_difference(Y, Z)))
% 190.93/24.63 = { by lemma 163 R->L }
% 190.93/24.63 domain_difference(domain_difference(Y, Z), c(X))
% 190.93/24.63 = { by lemma 162 R->L }
% 190.93/24.63 forward_diamond(domain(domain_difference(Y, Z)), X)
% 190.93/24.63 = { by lemma 133 }
% 190.93/24.63 forward_diamond(domain_difference(Y, Z), X)
% 190.93/24.63
% 190.93/24.63 Lemma 177: domain_difference(X, forward_box(c(Y), Z)) = domain_difference(domain_difference(X, Y), Z).
% 190.93/24.63 Proof:
% 190.93/24.63 domain_difference(X, forward_box(c(Y), Z))
% 190.93/24.63 = { by lemma 170 R->L }
% 190.93/24.63 multiplication(domain_difference(X, Y), c(Z))
% 190.93/24.63 = { by lemma 133 R->L }
% 190.93/24.63 multiplication(domain(domain_difference(X, Y)), c(Z))
% 190.93/24.63 = { by lemma 127 }
% 190.93/24.63 domain_difference(domain_difference(X, Y), Z)
% 190.93/24.63
% 190.93/24.63 Lemma 178: addition(c(X), domain_difference(X, c(Y))) = addition(c(X), domain(Y)).
% 190.93/24.63 Proof:
% 190.93/24.63 addition(c(X), domain_difference(X, c(Y)))
% 190.93/24.63 = { by lemma 163 R->L }
% 190.93/24.63 addition(c(X), domain_difference(Y, c(X)))
% 190.93/24.63 = { by lemma 68 R->L }
% 190.93/24.63 addition(domain(c(X)), domain_difference(Y, c(X)))
% 190.93/24.63 = { by lemma 160 }
% 190.93/24.63 addition(domain(c(X)), domain(Y))
% 190.93/24.63 = { by lemma 68 }
% 190.93/24.63 addition(c(X), domain(Y))
% 190.93/24.63
% 190.93/24.63 Lemma 179: multiplication(domain_difference(X, sK2_goals_X1), backward_box(sK3_goals_X0, sK1_goals_X2)) = domain_difference(X, sK2_goals_X1).
% 190.93/24.63 Proof:
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), backward_box(sK3_goals_X0, sK1_goals_X2))
% 190.93/24.63 = { by lemma 129 R->L }
% 190.93/24.63 multiplication(domain(X), multiplication(c(sK2_goals_X1), backward_box(sK3_goals_X0, sK1_goals_X2)))
% 190.93/24.63 = { by axiom 2 (complement) }
% 190.93/24.63 multiplication(domain(X), multiplication(antidomain(domain(sK2_goals_X1)), backward_box(sK3_goals_X0, sK1_goals_X2)))
% 190.93/24.63 = { by lemma 36 R->L }
% 190.93/24.63 multiplication(domain(X), multiplication(antidomain(domain(sK2_goals_X1)), addition(domain(sK2_goals_X1), backward_box(sK3_goals_X0, sK1_goals_X2))))
% 190.93/24.63 = { by axiom 5 (additive_commutativity) R->L }
% 190.93/24.63 multiplication(domain(X), multiplication(antidomain(domain(sK2_goals_X1)), addition(backward_box(sK3_goals_X0, sK1_goals_X2), domain(sK2_goals_X1))))
% 190.93/24.63 = { by lemma 150 R->L }
% 190.93/24.63 multiplication(domain(X), multiplication(antidomain(domain(sK2_goals_X1)), addition(backward_box(sK3_goals_X0, domain(sK1_goals_X2)), domain(sK2_goals_X1))))
% 190.93/24.63 = { by axiom 5 (additive_commutativity) R->L }
% 190.93/24.63 multiplication(domain(X), multiplication(antidomain(domain(sK2_goals_X1)), addition(domain(sK2_goals_X1), backward_box(sK3_goals_X0, domain(sK1_goals_X2)))))
% 190.93/24.63 = { by axiom 24 (goals) }
% 190.93/24.63 multiplication(domain(X), multiplication(antidomain(domain(sK2_goals_X1)), one))
% 190.93/24.63 = { by axiom 7 (multiplicative_right_identity) }
% 190.93/24.63 multiplication(domain(X), antidomain(domain(sK2_goals_X1)))
% 190.93/24.63 = { by axiom 2 (complement) R->L }
% 190.93/24.63 multiplication(domain(X), c(sK2_goals_X1))
% 190.93/24.63 = { by lemma 88 }
% 190.93/24.63 domain_difference(X, domain(sK2_goals_X1))
% 190.93/24.63 = { by lemma 89 }
% 190.93/24.63 domain_difference(X, sK2_goals_X1)
% 190.93/24.63
% 190.93/24.63 Lemma 180: domain(multiplication(X, multiplication(Y, domain(Z)))) = forward_diamond(multiplication(X, Y), Z).
% 190.93/24.63 Proof:
% 190.93/24.63 domain(multiplication(X, multiplication(Y, domain(Z))))
% 190.93/24.63 = { by axiom 21 (multiplicative_associativity) }
% 190.93/24.63 domain(multiplication(multiplication(X, Y), domain(Z)))
% 190.93/24.63 = { by axiom 18 (forward_diamond) R->L }
% 190.93/24.63 forward_diamond(multiplication(X, Y), Z)
% 190.93/24.63
% 190.93/24.63 Lemma 181: multiplication(X, multiplication(Y, codomain(multiplication(X, Y)))) = multiplication(X, Y).
% 190.93/24.63 Proof:
% 190.93/24.63 multiplication(X, multiplication(Y, codomain(multiplication(X, Y))))
% 190.93/24.63 = { by axiom 21 (multiplicative_associativity) }
% 190.93/24.63 multiplication(multiplication(X, Y), codomain(multiplication(X, Y)))
% 190.93/24.63 = { by lemma 53 }
% 190.93/24.63 multiplication(X, Y)
% 190.93/24.63
% 190.93/24.63 Lemma 182: forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(X, sK2_goals_X1)) = domain_difference(X, sK2_goals_X1).
% 190.93/24.63 Proof:
% 190.93/24.63 forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(X, sK2_goals_X1))
% 190.93/24.63 = { by lemma 172 R->L }
% 190.93/24.63 domain_difference(domain_difference(X, sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))
% 190.93/24.63 = { by lemma 154 R->L }
% 190.93/24.63 domain_difference(domain_difference(X, sK2_goals_X1), c(backward_box(sK3_goals_X0, sK1_goals_X2)))
% 190.93/24.63 = { by lemma 75 R->L }
% 190.93/24.63 multiplication(domain(domain_difference(X, sK2_goals_X1)), domain(backward_box(sK3_goals_X0, sK1_goals_X2)))
% 190.93/24.63 = { by lemma 133 }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), domain(backward_box(sK3_goals_X0, sK1_goals_X2)))
% 190.93/24.63 = { by lemma 76 R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), codomain(domain_difference(X, sK2_goals_X1))), domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1))))))
% 190.93/24.63 = { by axiom 5 (additive_commutativity) R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), codomain(domain_difference(X, sK2_goals_X1)))))
% 190.93/24.63 = { by lemma 166 R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), multiplication(domain(backward_box(sK3_goals_X0, sK1_goals_X2)), coantidomain(domain_difference(X, sK2_goals_X1)))))
% 190.93/24.63 = { by lemma 52 R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), multiplication(domain(backward_box(sK3_goals_X0, sK1_goals_X2)), coantidomain(domain_difference(X, sK2_goals_X1)))), coantidomain(domain_difference(X, sK2_goals_X1))))
% 190.93/24.63 = { by axiom 19 (additive_associativity) R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), addition(multiplication(domain(backward_box(sK3_goals_X0, sK1_goals_X2)), coantidomain(domain_difference(X, sK2_goals_X1))), coantidomain(domain_difference(X, sK2_goals_X1)))))
% 190.93/24.63 = { by axiom 5 (additive_commutativity) }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), addition(coantidomain(domain_difference(X, sK2_goals_X1)), multiplication(domain(backward_box(sK3_goals_X0, sK1_goals_X2)), coantidomain(domain_difference(X, sK2_goals_X1))))))
% 190.93/24.63 = { by lemma 61 R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), multiplication(addition(one, domain(backward_box(sK3_goals_X0, sK1_goals_X2))), coantidomain(domain_difference(X, sK2_goals_X1)))))
% 190.93/24.63 = { by lemma 65 }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), multiplication(one, coantidomain(domain_difference(X, sK2_goals_X1)))))
% 190.93/24.63 = { by axiom 9 (multiplicative_left_identity) }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), addition(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))), coantidomain(domain_difference(X, sK2_goals_X1))))
% 190.93/24.63 = { by lemma 52 }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(codomain(domain_difference(X, sK2_goals_X1)))))
% 190.93/24.63 = { by lemma 96 }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), coantidomain(domain_difference(X, sK2_goals_X1))))
% 190.93/24.63 = { by lemma 175 R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), codomain(domain_difference(X, sK2_goals_X1))))
% 190.93/24.63 = { by lemma 179 R->L }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), codomain(multiplication(domain_difference(X, sK2_goals_X1), backward_box(sK3_goals_X0, sK1_goals_X2)))))
% 190.93/24.63 = { by lemma 181 }
% 190.93/24.63 multiplication(domain_difference(X, sK2_goals_X1), backward_box(sK3_goals_X0, sK1_goals_X2))
% 190.93/24.63 = { by lemma 179 }
% 190.93/24.64 domain_difference(X, sK2_goals_X1)
% 190.93/24.64
% 190.93/24.64 Goal 1 (goals_1): addition(forward_box(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2)) = one.
% 190.93/24.64 Proof:
% 190.93/24.64 addition(forward_box(sK3_goals_X0, domain(sK2_goals_X1)), domain(sK1_goals_X2))
% 190.93/24.64 = { by lemma 149 }
% 190.93/24.64 addition(forward_box(sK3_goals_X0, sK2_goals_X1), domain(sK1_goals_X2))
% 190.93/24.64 = { by axiom 5 (additive_commutativity) }
% 190.93/24.64 addition(domain(sK1_goals_X2), forward_box(sK3_goals_X0, sK2_goals_X1))
% 190.93/24.64 = { by lemma 149 R->L }
% 190.93/24.64 addition(domain(sK1_goals_X2), forward_box(sK3_goals_X0, domain(sK2_goals_X1)))
% 190.93/24.64 = { by lemma 87 R->L }
% 190.93/24.64 addition(domain(sK1_goals_X2), forward_box(sK3_goals_X0, c(c(sK2_goals_X1))))
% 190.93/24.64 = { by lemma 87 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), forward_box(sK3_goals_X0, c(c(sK2_goals_X1))))
% 190.93/24.64 = { by axiom 5 (additive_commutativity) R->L }
% 190.93/24.64 addition(forward_box(sK3_goals_X0, c(c(sK2_goals_X1))), c(c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 139 R->L }
% 190.93/24.64 addition(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), c(c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 159 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), addition(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), domain_difference(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), c(sK1_goals_X2))))
% 190.93/24.64 = { by lemma 116 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), addition(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), multiplication(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), c(c(sK1_goals_X2)))))
% 190.93/24.64 = { by axiom 19 (additive_associativity) }
% 190.93/24.64 addition(addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), multiplication(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), c(c(sK1_goals_X2))))
% 190.93/24.64 = { by lemma 52 R->L }
% 190.93/24.64 addition(addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), multiplication(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.64 = { by lemma 60 R->L }
% 190.93/24.64 multiplication(addition(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), one), addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by axiom 5 (additive_commutativity) }
% 190.93/24.64 multiplication(addition(one, forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by axiom 18 (forward_diamond) }
% 190.93/24.64 multiplication(addition(one, domain(multiplication(sK3_goals_X0, domain(c(sK2_goals_X1))))), addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by lemma 65 }
% 190.93/24.64 multiplication(one, addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by axiom 9 (multiplicative_left_identity) }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.64 = { by lemma 70 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), backward_box(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), zero))
% 190.93/24.64 = { by lemma 64 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), c(codomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by lemma 67 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), c(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.64 = { by lemma 87 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by lemma 178 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.64 = { by lemma 163 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), c(c(sK1_goals_X2))))
% 190.93/24.64 = { by lemma 162 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), forward_diamond(domain(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 110 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), forward_diamond(domain_difference(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 147 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), forward_diamond(domain_difference(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 176 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), forward_box(domain(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 168 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain(c(sK1_goals_X2)), domain_difference(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by axiom 20 (domain_difference) }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain(c(sK1_goals_X2)), multiplication(domain(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), antidomain(codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))))
% 190.93/24.64 = { by lemma 117 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), antidomain(codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 37 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), c(codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 170 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), forward_box(c(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 177 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 166 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), coantidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 97 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain_difference(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), zero), coantidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 51 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(addition(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), domain_difference(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), zero)), coantidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 97 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(addition(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), domain(domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), coantidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 133 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(addition(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), domain_difference(c(sK1_goals_X2), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), coantidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 76 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), multiplication(domain(c(sK1_goals_X2)), coantidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 166 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), codomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 147 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.64 = { by lemma 89 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 87 R->L }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), c(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 178 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), domain(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.64 = { by lemma 68 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.64 = { by lemma 109 }
% 190.93/24.64 addition(c(c(sK1_goals_X2)), forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.64 = { by axiom 5 (additive_commutativity) R->L }
% 190.93/24.64 addition(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 109 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), c(c(sK1_goals_X2)))
% 190.93/24.64 = { by lemma 68 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), domain(c(c(sK1_goals_X2))))
% 190.93/24.64 = { by lemma 76 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(domain_difference(c(c(sK1_goals_X2)), forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 109 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(domain_difference(c(c(sK1_goals_X2)), c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 37 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(domain_difference(c(c(sK1_goals_X2)), antidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 37 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(domain_difference(antidomain(c(sK1_goals_X2)), antidomain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 79 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 37 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(antidomain(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by axiom 9 (multiplicative_left_identity) R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(multiplication(one, antidomain(c(sK1_goals_X2))), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 39 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(multiplication(domain(one), antidomain(c(sK1_goals_X2))), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by axiom 20 (domain_difference) R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(domain_difference(one, c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 45 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(domain_difference(c(zero), c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 37 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(domain_difference(antidomain(zero), c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 129 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(domain(antidomain(zero)), multiplication(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 31 }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(c(zero), multiplication(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by lemma 119 R->L }
% 190.93/24.64 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(c(multiplication(antidomain(domain(c(sK1_goals_X2))), multiplication(domain(c(sK1_goals_X2)), antidomain(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), multiplication(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.64 = { by axiom 20 (domain_difference) R->L }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(c(multiplication(antidomain(domain(c(sK1_goals_X2))), domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), multiplication(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.65 = { by axiom 2 (complement) R->L }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(multiplication(c(multiplication(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), multiplication(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.65 = { by lemma 56 }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), addition(zero, domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.65 = { by lemma 46 }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), domain_difference(c(c(sK1_goals_X2)), c(forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.65 = { by lemma 106 }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), domain_difference(c(c(sK1_goals_X2)), forward_diamond(domain(c(sK1_goals_X2)), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.65 = { by lemma 101 }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), domain_difference(c(c(sK1_goals_X2)), domain(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.65 = { by lemma 89 }
% 190.93/24.65 addition(c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), domain_difference(c(c(sK1_goals_X2)), domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.65 = { by lemma 107 }
% 190.93/24.65 c(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.65 = { by lemma 109 }
% 190.93/24.65 forward_box(domain(c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by axiom 16 (forward_box) }
% 190.93/24.65 c(forward_diamond(domain(c(sK1_goals_X2)), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.65 = { by axiom 18 (forward_diamond) }
% 190.93/24.65 c(domain(multiplication(domain(c(sK1_goals_X2)), domain(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))))
% 190.93/24.65 = { by lemma 34 R->L }
% 190.93/24.65 c(domain(multiplication(domain(c(sK1_goals_X2)), multiplication(domain(domain(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), domain(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))))
% 190.93/24.65 = { by lemma 180 }
% 190.93/24.65 c(forward_diamond(multiplication(domain(c(sK1_goals_X2)), domain(domain(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.65 = { by lemma 49 }
% 190.93/24.65 c(forward_diamond(multiplication(domain(c(sK1_goals_X2)), forward_diamond(one, c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.65 = { by lemma 81 }
% 190.93/24.65 c(forward_diamond(multiplication(domain(c(sK1_goals_X2)), domain(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))))
% 190.93/24.65 = { by axiom 16 (forward_box) R->L }
% 190.93/24.65 forward_box(multiplication(domain(c(sK1_goals_X2)), domain(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by lemma 68 }
% 190.93/24.65 forward_box(multiplication(domain(c(sK1_goals_X2)), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by lemma 88 }
% 190.93/24.65 forward_box(domain_difference(c(sK1_goals_X2), domain(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by lemma 89 }
% 190.93/24.65 forward_box(domain_difference(c(sK1_goals_X2), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by lemma 161 R->L }
% 190.93/24.65 forward_box(forward_diamond(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by lemma 126 R->L }
% 190.93/24.65 forward_box(domain(multiplication(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2))), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))))
% 190.93/24.65 = { by lemma 109 R->L }
% 190.93/24.65 c(domain_difference(multiplication(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))
% 190.93/24.65 = { by axiom 6 (additive_identity) R->L }
% 190.93/24.65 c(addition(domain_difference(multiplication(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), zero))
% 190.93/24.65 = { by lemma 128 R->L }
% 190.93/24.65 c(addition(domain_difference(multiplication(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)), coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), domain_difference(multiplication(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)), c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))))))
% 190.93/24.65 = { by lemma 76 }
% 190.93/24.65 c(domain(multiplication(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2))))
% 190.93/24.65 = { by lemma 126 }
% 190.93/24.65 c(forward_diamond(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(sK1_goals_X2)))
% 190.93/24.65 = { by lemma 122 }
% 190.93/24.65 forward_box(c(coantidomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)))), c(c(sK1_goals_X2)))
% 190.93/24.65 = { by lemma 67 }
% 190.93/24.65 forward_box(codomain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), c(c(sK1_goals_X2)))
% 190.93/24.65 = { by lemma 138 }
% 190.93/24.65 forward_box(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), c(c(sK1_goals_X2)))
% 190.93/24.65 = { by lemma 87 }
% 190.93/24.65 forward_box(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), domain(sK1_goals_X2))
% 190.93/24.65 = { by lemma 149 }
% 190.93/24.65 forward_box(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), sK1_goals_X2)
% 190.93/24.65 = { by lemma 113 R->L }
% 190.93/24.65 forward_box(domain(forward_diamond(sK3_goals_X0, c(sK2_goals_X1))), sK1_goals_X2)
% 190.93/24.65 = { by lemma 109 R->L }
% 190.93/24.65 c(domain_difference(forward_diamond(sK3_goals_X0, c(sK2_goals_X1)), sK1_goals_X2))
% 190.93/24.65 = { by lemma 164 R->L }
% 190.93/24.65 c(domain_difference(forward_diamond(sK3_goals_X0, domain_difference(c(sK2_goals_X1), sK2_goals_X1)), sK1_goals_X2))
% 190.93/24.65 = { by lemma 161 R->L }
% 190.93/24.65 c(domain_difference(forward_diamond(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 153 R->L }
% 190.93/24.65 c(domain_difference(forward_diamond(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), domain(c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 87 R->L }
% 190.93/24.65 c(domain_difference(forward_diamond(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(c(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 106 R->L }
% 190.93/24.65 c(domain_difference(forward_diamond(sK3_goals_X0, c(forward_box(c(sK2_goals_X1), c(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 106 R->L }
% 190.93/24.65 c(domain_difference(c(forward_box(sK3_goals_X0, forward_box(c(sK2_goals_X1), c(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 37 R->L }
% 190.93/24.65 c(domain_difference(antidomain(forward_box(sK3_goals_X0, forward_box(c(sK2_goals_X1), c(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 122 R->L }
% 190.93/24.65 c(domain_difference(antidomain(forward_box(sK3_goals_X0, c(forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 37 R->L }
% 190.93/24.65 c(domain_difference(antidomain(forward_box(sK3_goals_X0, antidomain(forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 104 R->L }
% 190.93/24.65 c(domain_difference(antidomain(forward_box(sK3_goals_X0, c(multiplication(c(sK2_goals_X1), domain(c(sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 122 R->L }
% 190.93/24.65 c(domain_difference(antidomain(c(forward_diamond(sK3_goals_X0, multiplication(c(sK2_goals_X1), domain(c(sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 37 R->L }
% 190.93/24.65 c(domain_difference(antidomain(antidomain(forward_diamond(sK3_goals_X0, multiplication(c(sK2_goals_X1), domain(c(sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by axiom 18 (forward_diamond) }
% 190.93/24.65 c(domain_difference(antidomain(antidomain(domain(multiplication(sK3_goals_X0, domain(multiplication(c(sK2_goals_X1), domain(c(sK2_goals_X1)))))))), sK1_goals_X2))
% 190.93/24.65 = { by axiom 18 (forward_diamond) R->L }
% 190.93/24.65 c(domain_difference(antidomain(antidomain(domain(multiplication(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by axiom 2 (complement) R->L }
% 190.93/24.65 c(domain_difference(antidomain(c(multiplication(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 47 }
% 190.93/24.65 c(domain_difference(domain(domain(multiplication(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 49 }
% 190.93/24.65 c(domain_difference(forward_diamond(one, multiplication(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 81 }
% 190.93/24.65 c(domain_difference(domain(multiplication(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 167 R->L }
% 190.93/24.65 c(domain_difference(domain(multiplication(sK3_goals_X0, domain_difference(c(sK2_goals_X1), c(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 37 R->L }
% 190.93/24.65 c(domain_difference(domain(multiplication(sK3_goals_X0, domain_difference(c(sK2_goals_X1), antidomain(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 78 R->L }
% 190.93/24.65 c(domain_difference(domain(multiplication(sK3_goals_X0, multiplication(domain(c(sK2_goals_X1)), domain(c(sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 180 }
% 190.93/24.65 c(domain_difference(forward_diamond(multiplication(sK3_goals_X0, domain(c(sK2_goals_X1))), c(sK2_goals_X1)), sK1_goals_X2))
% 190.93/24.65 = { by lemma 68 }
% 190.93/24.65 c(domain_difference(forward_diamond(multiplication(sK3_goals_X0, c(sK2_goals_X1)), c(sK2_goals_X1)), sK1_goals_X2))
% 190.93/24.65 = { by lemma 116 R->L }
% 190.93/24.65 c(multiplication(forward_diamond(multiplication(sK3_goals_X0, c(sK2_goals_X1)), c(sK2_goals_X1)), c(sK1_goals_X2)))
% 190.93/24.65 = { by lemma 180 R->L }
% 190.93/24.65 c(multiplication(domain(multiplication(sK3_goals_X0, multiplication(c(sK2_goals_X1), domain(c(sK2_goals_X1))))), c(sK1_goals_X2)))
% 190.93/24.65 = { by lemma 127 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(c(sK2_goals_X1), domain(c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 31 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(domain(antidomain(sK2_goals_X1)), domain(c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 78 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, domain_difference(antidomain(sK2_goals_X1), antidomain(c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 37 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, domain_difference(c(sK2_goals_X1), antidomain(c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 37 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, domain_difference(c(sK2_goals_X1), c(c(sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 167 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, forward_diamond(c(sK2_goals_X1), c(sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 161 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, domain_difference(c(sK2_goals_X1), sK2_goals_X1)), sK1_goals_X2))
% 190.93/24.65 = { by lemma 182 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 173 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 181 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), codomain(multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 173 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), codomain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 98 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(forward_diamond(domain_difference(c(sK2_goals_X1), sK2_goals_X1), one), codomain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 171 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(forward_diamond(domain_difference(c(sK2_goals_X1), sK2_goals_X1), one), coantidomain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 98 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), coantidomain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 132 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), coantidomain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 139 }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), forward_box(backward_box(sK3_goals_X0, sK1_goals_X2), c(domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 122 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), c(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 163 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)), c(domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 114 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1))), c(domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 162 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain(multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by axiom 18 (forward_diamond) R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 113 R->L }
% 190.93/24.65 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.65 = { by lemma 155 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain(domain_difference(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)), c(backward_box(sK3_goals_X0, sK1_goals_X2)))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 124 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(forward_diamond(domain(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), backward_box(sK3_goals_X0, sK1_goals_X2)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 113 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(forward_diamond(forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)), backward_box(sK3_goals_X0, sK1_goals_X2)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 172 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(forward_diamond(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2))), backward_box(sK3_goals_X0, sK1_goals_X2)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 176 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), forward_box(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 109 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), c(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2))))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 141 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), forward_box(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2))), coantidomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 147 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), forward_box(codomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))), coantidomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 142 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), coantidomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2))))), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 176 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), forward_box(domain(backward_box(sK3_goals_X0, sK1_goals_X2)), coantidomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))))))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 168 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), domain_difference(backward_box(sK3_goals_X0, sK1_goals_X2), coantidomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))))))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 175 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), codomain(domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2)))))))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 165 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), codomain(multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), backward_box(sK3_goals_X0, sK1_goals_X2))))))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 181 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), multiplication(domain(domain_difference(c(sK2_goals_X1), sK2_goals_X1)), backward_box(sK3_goals_X0, sK1_goals_X2)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 165 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(domain_difference(c(sK2_goals_X1), sK2_goals_X1), backward_diamond(sK3_goals_X0, c(sK1_goals_X2))))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 172 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), forward_diamond(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 182 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, sK1_goals_X2), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 151 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, backward_box(one, sK1_goals_X2)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 74 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(backward_box(sK3_goals_X0, coantidomain(c(sK1_goals_X2))), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 145 R->L }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(backward_diamond(sK3_goals_X0, c(sK1_goals_X2))), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.66 = { by axiom 15 (backward_diamond) }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(codomain(multiplication(codomain(c(sK1_goals_X2)), sK3_goals_X0))), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 50 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(codomain(coantidomain(multiplication(codomain(c(sK1_goals_X2)), sK3_goals_X0))), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 190.93/24.66 = { by lemma 54 }
% 190.93/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(codomain(c(sK1_goals_X2)), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 69 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(domain(codomain(c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 76 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(domain_difference(codomain(c(sK1_goals_X2)), sK1_goals_X2), domain_difference(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 89 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(domain_difference(codomain(c(sK1_goals_X2)), domain(sK1_goals_X2)), domain_difference(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 88 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(multiplication(domain(codomain(c(sK1_goals_X2))), c(sK1_goals_X2)), domain_difference(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 169 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(multiplication(domain(codomain(c(sK1_goals_X2))), multiplication(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), domain_difference(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 86 }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(multiplication(codomain(c(sK1_goals_X2)), c(sK1_goals_X2)), domain_difference(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 169 }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), domain_difference(codomain(c(sK1_goals_X2)), c(sK1_goals_X2))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 91 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), c(c(sK1_goals_X2)))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 164 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 92 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by axiom 9 (multiplicative_left_identity) R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(one, backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 44 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(zero), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by axiom 12 (domain1) R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), domain(sK1_goals_X2))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by axiom 9 (multiplicative_left_identity) R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), multiplication(one, domain(sK1_goals_X2)))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 65 R->L }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), multiplication(addition(one, domain(c(c(sK1_goals_X2)))), domain(sK1_goals_X2)))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 61 }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), addition(domain(sK1_goals_X2), multiplication(domain(c(c(sK1_goals_X2))), domain(sK1_goals_X2))))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 78 }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), addition(domain(sK1_goals_X2), domain_difference(c(c(sK1_goals_X2)), antidomain(sK1_goals_X2))))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 37 }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), addition(domain(sK1_goals_X2), domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2))))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.66 = { by lemma 36 }
% 191.27/24.66 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(antidomain(domain(sK1_goals_X2)), domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.67 = { by axiom 2 (complement) R->L }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), multiplication(coantidomain(multiplication(c(sK1_goals_X2), domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)))), backward_diamond(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), c(sK1_goals_X2)))))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.67 = { by lemma 93 }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), multiplication(domain_difference(c(c(sK1_goals_X2)), c(sK1_goals_X2)), zero))), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.67 = { by axiom 8 (right_annihilation) }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), multiplication(codomain(c(sK1_goals_X2)), zero)), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.67 = { by axiom 8 (right_annihilation) }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(addition(c(sK1_goals_X2), zero), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.67 = { by axiom 6 (additive_identity) }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2))
% 191.27/24.67 = { by lemma 133 R->L }
% 191.27/24.67 c(domain(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2)))
% 191.27/24.67 = { by lemma 97 R->L }
% 191.27/24.67 c(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2), zero))
% 191.27/24.67 = { by axiom 10 (left_annihilation) R->L }
% 191.27/24.67 c(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2), multiplication(zero, domain_difference(c(sK2_goals_X1), sK2_goals_X1))))
% 191.27/24.67 = { by axiom 11 (codomain1) R->L }
% 191.27/24.67 c(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2), multiplication(multiplication(multiplication(c(sK1_goals_X2), sK3_goals_X0), coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0))), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))
% 191.27/24.67 = { by axiom 21 (multiplicative_associativity) R->L }
% 191.27/24.67 c(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2), multiplication(multiplication(c(sK1_goals_X2), sK3_goals_X0), multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by axiom 21 (multiplicative_associativity) R->L }
% 191.27/24.67 c(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))
% 191.27/24.67 = { by lemma 89 R->L }
% 191.27/24.67 c(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), domain(sK1_goals_X2)), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))
% 191.27/24.67 = { by lemma 133 R->L }
% 191.27/24.67 c(domain(domain_difference(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), domain(sK1_goals_X2)), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 101 R->L }
% 191.27/24.67 c(forward_diamond(domain(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), domain(sK1_goals_X2))), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 133 }
% 191.27/24.67 c(forward_diamond(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), domain(sK1_goals_X2)), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 87 R->L }
% 191.27/24.67 c(forward_diamond(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), c(c(sK1_goals_X2))), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 75 R->L }
% 191.27/24.67 c(forward_diamond(multiplication(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), domain(c(sK1_goals_X2))), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 37 R->L }
% 191.27/24.67 c(forward_diamond(multiplication(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), domain(c(sK1_goals_X2))), antidomain(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 180 R->L }
% 191.27/24.67 c(domain(multiplication(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), multiplication(domain(c(sK1_goals_X2)), domain(antidomain(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))))))
% 191.27/24.67 = { by lemma 31 }
% 191.27/24.67 c(domain(multiplication(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), multiplication(domain(c(sK1_goals_X2)), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))))
% 191.27/24.67 = { by lemma 88 }
% 191.27/24.67 c(domain(multiplication(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), domain_difference(c(sK1_goals_X2), domain(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))))
% 191.27/24.67 = { by lemma 89 }
% 191.27/24.67 c(domain(multiplication(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), domain_difference(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))))
% 191.27/24.67 = { by lemma 125 }
% 191.27/24.67 c(forward_diamond(domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))), domain_difference(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 162 }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), c(domain_difference(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))))
% 191.27/24.67 = { by lemma 109 }
% 191.27/24.67 c(domain_difference(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))), forward_box(domain(c(sK1_goals_X2)), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))
% 191.27/24.67 = { by lemma 176 }
% 191.27/24.67 c(forward_diamond(domain_difference(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 130 R->L }
% 191.27/24.67 c(forward_diamond(forward_diamond(c(sK1_goals_X2), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 106 R->L }
% 191.27/24.67 c(forward_diamond(c(forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 37 R->L }
% 191.27/24.67 c(forward_diamond(antidomain(forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 35 R->L }
% 191.27/24.67 c(forward_diamond(multiplication(c(forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), antidomain(forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 106 }
% 191.27/24.67 c(forward_diamond(multiplication(forward_diamond(c(sK1_goals_X2), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), antidomain(forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 37 }
% 191.27/24.67 c(forward_diamond(multiplication(forward_diamond(c(sK1_goals_X2), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), c(forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 112 }
% 191.27/24.67 c(forward_diamond(domain_difference(multiplication(c(sK1_goals_X2), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 68 }
% 191.27/24.67 c(forward_diamond(domain_difference(multiplication(c(sK1_goals_X2), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 115 }
% 191.27/24.67 c(forward_diamond(domain_difference(forward_diamond(c(sK1_goals_X2), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 161 }
% 191.27/24.67 c(forward_diamond(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), forward_box(c(sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 177 }
% 191.27/24.67 c(forward_diamond(domain_difference(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 111 }
% 191.27/24.67 c(forward_diamond(domain_difference(domain(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2)), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by lemma 132 }
% 191.27/24.67 c(forward_diamond(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.67 = { by axiom 6 (additive_identity) R->L }
% 191.27/24.67 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), zero), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 128 R->L }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(multiplication(domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2)), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 81 R->L }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(multiplication(forward_diamond(one, c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2)), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 49 R->L }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(multiplication(domain(domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2)), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by axiom 20 (domain_difference) }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(multiplication(domain(domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), multiplication(domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), antidomain(sK1_goals_X2))), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 86 }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(multiplication(domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))), antidomain(sK1_goals_X2)), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by axiom 20 (domain_difference) R->L }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), domain(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 89 }
% 191.27/24.68 c(forward_diamond(addition(domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), domain_difference(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 76 }
% 191.27/24.68 c(forward_diamond(domain(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 102 }
% 191.27/24.68 c(forward_diamond(forward_diamond(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), c(sK1_goals_X2)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 130 }
% 191.27/24.68 c(forward_diamond(domain_difference(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 90 R->L }
% 191.27/24.68 c(forward_diamond(multiplication(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), c(sK1_goals_X2)), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1)))))
% 191.27/24.68 = { by lemma 180 R->L }
% 191.27/24.68 c(domain(multiplication(c(multiplication(c(sK1_goals_X2), multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))), multiplication(c(sK1_goals_X2), domain(multiplication(sK3_goals_X0, multiplication(coantidomain(multiplication(c(sK1_goals_X2), sK3_goals_X0)), domain_difference(c(sK2_goals_X1), sK2_goals_X1))))))))
% 191.27/24.68 = { by lemma 56 }
% 191.27/24.68 c(domain(zero))
% 191.27/24.68 = { by lemma 38 R->L }
% 191.27/24.68 c(domain(antidomain(one)))
% 191.27/24.68 = { by lemma 31 }
% 191.27/24.68 c(c(one))
% 191.27/24.68 = { by axiom 2 (complement) }
% 191.27/24.68 c(antidomain(domain(one)))
% 191.27/24.68 = { by lemma 39 }
% 191.27/24.68 c(antidomain(one))
% 191.27/24.68 = { by lemma 38 }
% 191.27/24.68 c(zero)
% 191.27/24.68 = { by lemma 45 }
% 191.27/24.68 one
% 191.27/24.68 % SZS output end Proof
% 191.27/24.68
% 191.27/24.68 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------