TSTP Solution File: GRP655+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP655+1 : TPTP v8.1.2. Released v4.0.0.
% 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 : n002.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 01:19:29 EDT 2023
% Result : Theorem 6.51s 1.17s
% Output : Proof 6.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP655+1 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Mon Aug 28 20:08:04 EDT 2023
% 0.12/0.34 % CPUTime :
% 6.51/1.17 Command-line arguments: --ground-connectedness --complete-subsets
% 6.51/1.17
% 6.51/1.17 % SZS status Theorem
% 6.51/1.17
% 6.56/1.24 % SZS output start Proof
% 6.56/1.24 Take the following subset of the input axioms:
% 6.56/1.24 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 6.56/1.24 fof(f02, axiom, ![B2, A2]: ld(A2, mult(A2, B2))=B2).
% 6.56/1.24 fof(f03, axiom, ![B2, A2]: mult(rd(A2, B2), B2)=A2).
% 6.56/1.24 fof(f04, axiom, ![B2, A2]: rd(mult(A2, B2), B2)=A2).
% 6.56/1.24 fof(f05, axiom, ![C, B2, A2]: mult(A2, mult(B2, mult(C, B2)))=mult(mult(mult(A2, B2), C), B2)).
% 6.56/1.24 fof(goals, conjecture, ?[X0]: ![X1]: (mult(X1, X0)=X1 & mult(X0, X1)=X1)).
% 6.56/1.24
% 6.56/1.24 Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.56/1.24 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.56/1.24 We repeatedly replace C & s=t => u=v by the two clauses:
% 6.56/1.24 fresh(y, y, x1...xn) = u
% 6.56/1.24 C => fresh(s, t, x1...xn) = v
% 6.56/1.24 where fresh is a fresh function symbol and x1..xn are the free
% 6.56/1.24 variables of u and v.
% 6.56/1.24 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.56/1.24 input problem has no model of domain size 1).
% 6.56/1.24
% 6.56/1.24 The encoding turns the above axioms into the following unit equations and goals:
% 6.56/1.24
% 6.56/1.24 Axiom 1 (f02): ld(X, mult(X, Y)) = Y.
% 6.56/1.24 Axiom 2 (f04): rd(mult(X, Y), Y) = X.
% 6.56/1.24 Axiom 3 (f01): mult(X, ld(X, Y)) = Y.
% 6.56/1.24 Axiom 4 (f03): mult(rd(X, Y), Y) = X.
% 6.56/1.24 Axiom 5 (f05): mult(X, mult(Y, mult(Z, Y))) = mult(mult(mult(X, Y), Z), Y).
% 6.56/1.24
% 6.56/1.24 Lemma 6: rd(X, ld(Y, X)) = Y.
% 6.56/1.24 Proof:
% 6.56/1.24 rd(X, ld(Y, X))
% 6.56/1.24 = { by axiom 3 (f01) R->L }
% 6.56/1.24 rd(mult(Y, ld(Y, X)), ld(Y, X))
% 6.56/1.24 = { by axiom 2 (f04) }
% 6.56/1.24 Y
% 6.56/1.24
% 6.56/1.24 Lemma 7: mult(rd(X, Y), mult(Y, mult(Z, Y))) = mult(mult(X, Z), Y).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(rd(X, Y), mult(Y, mult(Z, Y)))
% 6.56/1.24 = { by axiom 5 (f05) }
% 6.56/1.24 mult(mult(mult(rd(X, Y), Y), Z), Y)
% 6.56/1.24 = { by axiom 4 (f03) }
% 6.56/1.24 mult(mult(X, Z), Y)
% 6.56/1.24
% 6.56/1.24 Lemma 8: ld(rd(X, Y), mult(mult(X, Z), Y)) = mult(Y, mult(Z, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 ld(rd(X, Y), mult(mult(X, Z), Y))
% 6.56/1.24 = { by lemma 7 R->L }
% 6.56/1.24 ld(rd(X, Y), mult(rd(X, Y), mult(Y, mult(Z, Y))))
% 6.56/1.24 = { by axiom 1 (f02) }
% 6.56/1.24 mult(Y, mult(Z, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 9: mult(X, mult(ld(Y, Z), X)) = ld(rd(Y, X), mult(Z, X)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(X, mult(ld(Y, Z), X))
% 6.56/1.24 = { by lemma 8 R->L }
% 6.56/1.24 ld(rd(Y, X), mult(mult(Y, ld(Y, Z)), X))
% 6.56/1.24 = { by axiom 3 (f01) }
% 6.56/1.24 ld(rd(Y, X), mult(Z, X))
% 6.56/1.24
% 6.56/1.24 Lemma 10: ld(X, ld(rd(Y, X), mult(Z, X))) = mult(ld(Y, Z), X).
% 6.56/1.24 Proof:
% 6.56/1.24 ld(X, ld(rd(Y, X), mult(Z, X)))
% 6.56/1.24 = { by lemma 9 R->L }
% 6.56/1.24 ld(X, mult(X, mult(ld(Y, Z), X)))
% 6.56/1.24 = { by axiom 1 (f02) }
% 6.56/1.24 mult(ld(Y, Z), X)
% 6.56/1.24
% 6.56/1.24 Lemma 11: mult(ld(X, rd(X, Y)), Y) = ld(Y, Y).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(ld(X, rd(X, Y)), Y)
% 6.56/1.24 = { by lemma 10 R->L }
% 6.56/1.24 ld(Y, ld(rd(X, Y), mult(rd(X, Y), Y)))
% 6.56/1.24 = { by axiom 1 (f02) }
% 6.56/1.24 ld(Y, Y)
% 6.56/1.24
% 6.56/1.24 Lemma 12: mult(ld(X, Y), ld(Y, X)) = ld(ld(Y, X), ld(Y, X)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(ld(X, Y), ld(Y, X))
% 6.56/1.24 = { by lemma 6 R->L }
% 6.56/1.24 mult(ld(X, rd(X, ld(Y, X))), ld(Y, X))
% 6.56/1.24 = { by lemma 11 }
% 6.56/1.24 ld(ld(Y, X), ld(Y, X))
% 6.56/1.24
% 6.56/1.24 Lemma 13: ld(rd(X, Y), X) = Y.
% 6.56/1.24 Proof:
% 6.56/1.24 ld(rd(X, Y), X)
% 6.56/1.24 = { by axiom 4 (f03) R->L }
% 6.56/1.24 ld(rd(X, Y), mult(rd(X, Y), Y))
% 6.56/1.24 = { by axiom 1 (f02) }
% 6.56/1.24 Y
% 6.56/1.24
% 6.56/1.24 Lemma 14: mult(ld(mult(X, Y), X), Y) = ld(Y, Y).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(ld(mult(X, Y), X), Y)
% 6.56/1.24 = { by lemma 10 R->L }
% 6.56/1.24 ld(Y, ld(rd(mult(X, Y), Y), mult(X, Y)))
% 6.56/1.24 = { by lemma 13 }
% 6.56/1.24 ld(Y, Y)
% 6.56/1.24
% 6.56/1.24 Lemma 15: rd(ld(X, X), X) = ld(Y, rd(Y, X)).
% 6.56/1.24 Proof:
% 6.56/1.24 rd(ld(X, X), X)
% 6.56/1.24 = { by lemma 11 R->L }
% 6.56/1.24 rd(mult(ld(Y, rd(Y, X)), X), X)
% 6.56/1.24 = { by axiom 2 (f04) }
% 6.56/1.24 ld(Y, rd(Y, X))
% 6.56/1.24
% 6.56/1.24 Lemma 16: ld(mult(X, Y), X) = ld(Z, rd(Z, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 ld(mult(X, Y), X)
% 6.56/1.24 = { by axiom 2 (f04) R->L }
% 6.56/1.24 rd(mult(ld(mult(X, Y), X), Y), Y)
% 6.56/1.24 = { by lemma 14 }
% 6.56/1.24 rd(ld(Y, Y), Y)
% 6.56/1.24 = { by lemma 15 }
% 6.56/1.24 ld(Z, rd(Z, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 17: mult(X, mult(ld(X, Y), mult(Z, ld(X, Y)))) = mult(mult(Y, Z), ld(X, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(X, mult(ld(X, Y), mult(Z, ld(X, Y))))
% 6.56/1.24 = { by axiom 5 (f05) }
% 6.56/1.24 mult(mult(mult(X, ld(X, Y)), Z), ld(X, Y))
% 6.56/1.24 = { by axiom 3 (f01) }
% 6.56/1.24 mult(mult(Y, Z), ld(X, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 18: mult(mult(X, Y), ld(Y, X)) = mult(Y, mult(ld(Y, X), X)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(mult(X, Y), ld(Y, X))
% 6.56/1.24 = { by lemma 17 R->L }
% 6.56/1.24 mult(Y, mult(ld(Y, X), mult(Y, ld(Y, X))))
% 6.56/1.24 = { by axiom 3 (f01) }
% 6.56/1.24 mult(Y, mult(ld(Y, X), X))
% 6.56/1.24
% 6.56/1.24 Lemma 19: ld(Z, rd(Z, Y)) = ld(X, rd(X, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 ld(Z, rd(Z, Y))
% 6.56/1.24 = { by axiom 2 (f04) R->L }
% 6.56/1.24 rd(mult(ld(Z, rd(Z, Y)), Y), Y)
% 6.56/1.24 = { by lemma 11 }
% 6.56/1.24 rd(ld(Y, Y), Y)
% 6.56/1.24 = { by lemma 11 R->L }
% 6.56/1.24 rd(mult(ld(X, rd(X, Y)), Y), Y)
% 6.56/1.24 = { by axiom 2 (f04) }
% 6.56/1.24 ld(X, rd(X, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 20: ld(X, rd(X, ld(Y, Z))) = ld(Z, Y).
% 6.56/1.24 Proof:
% 6.56/1.24 ld(X, rd(X, ld(Y, Z)))
% 6.56/1.24 = { by lemma 19 }
% 6.56/1.24 ld(Z, rd(Z, ld(Y, Z)))
% 6.56/1.24 = { by lemma 6 }
% 6.56/1.24 ld(Z, Y)
% 6.56/1.24
% 6.56/1.24 Lemma 21: rd(mult(X, mult(Y, mult(Z, Y))), Y) = mult(mult(X, Y), Z).
% 6.56/1.24 Proof:
% 6.56/1.24 rd(mult(X, mult(Y, mult(Z, Y))), Y)
% 6.56/1.24 = { by axiom 5 (f05) }
% 6.56/1.24 rd(mult(mult(mult(X, Y), Z), Y), Y)
% 6.56/1.24 = { by axiom 2 (f04) }
% 6.56/1.24 mult(mult(X, Y), Z)
% 6.56/1.24
% 6.56/1.24 Lemma 22: mult(mult(X, Y), rd(Z, Y)) = rd(mult(X, mult(Y, Z)), Y).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(mult(X, Y), rd(Z, Y))
% 6.56/1.24 = { by lemma 21 R->L }
% 6.56/1.24 rd(mult(X, mult(Y, mult(rd(Z, Y), Y))), Y)
% 6.56/1.24 = { by axiom 4 (f03) }
% 6.56/1.24 rd(mult(X, mult(Y, Z)), Y)
% 6.56/1.24
% 6.56/1.24 Lemma 23: rd(mult(rd(X, Y), mult(Y, Z)), Y) = mult(X, rd(Z, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 rd(mult(rd(X, Y), mult(Y, Z)), Y)
% 6.56/1.24 = { by lemma 22 R->L }
% 6.56/1.24 mult(mult(rd(X, Y), Y), rd(Z, Y))
% 6.56/1.24 = { by axiom 4 (f03) }
% 6.56/1.24 mult(X, rd(Z, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 24: mult(X, ld(Y, rd(Y, Z))) = rd(X, Z).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(X, ld(Y, rd(Y, Z)))
% 6.56/1.24 = { by lemma 15 R->L }
% 6.56/1.24 mult(X, rd(ld(Z, Z), Z))
% 6.56/1.24 = { by lemma 23 R->L }
% 6.56/1.24 rd(mult(rd(X, Z), mult(Z, ld(Z, Z))), Z)
% 6.56/1.24 = { by axiom 3 (f01) }
% 6.56/1.24 rd(mult(rd(X, Z), Z), Z)
% 6.56/1.24 = { by axiom 2 (f04) }
% 6.56/1.24 rd(X, Z)
% 6.56/1.24
% 6.56/1.24 Lemma 25: mult(X, ld(Y, Z)) = rd(X, ld(Z, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(X, ld(Y, Z))
% 6.56/1.24 = { by lemma 20 R->L }
% 6.56/1.24 mult(X, ld(W, rd(W, ld(Z, Y))))
% 6.56/1.24 = { by lemma 24 }
% 6.56/1.24 rd(X, ld(Z, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 26: mult(X, mult(ld(X, Y), Y)) = rd(mult(Y, X), ld(Y, X)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(X, mult(ld(X, Y), Y))
% 6.56/1.24 = { by axiom 1 (f02) R->L }
% 6.56/1.24 mult(X, mult(ld(ld(Y, mult(Y, X)), Y), Y))
% 6.56/1.24 = { by axiom 1 (f02) R->L }
% 6.56/1.24 mult(ld(Y, mult(Y, X)), mult(ld(ld(Y, mult(Y, X)), Y), Y))
% 6.56/1.24 = { by lemma 18 R->L }
% 6.56/1.24 mult(mult(Y, ld(Y, mult(Y, X))), ld(ld(Y, mult(Y, X)), Y))
% 6.56/1.24 = { by axiom 3 (f01) }
% 6.56/1.24 mult(mult(Y, X), ld(ld(Y, mult(Y, X)), Y))
% 6.56/1.24 = { by lemma 25 }
% 6.56/1.24 rd(mult(Y, X), ld(Y, ld(Y, mult(Y, X))))
% 6.56/1.24 = { by axiom 1 (f02) }
% 6.56/1.24 rd(mult(Y, X), ld(Y, X))
% 6.56/1.24
% 6.56/1.24 Lemma 27: rd(X, ld(Y, rd(Y, Z))) = mult(X, Z).
% 6.56/1.24 Proof:
% 6.56/1.24 rd(X, ld(Y, rd(Y, Z)))
% 6.56/1.24 = { by lemma 16 R->L }
% 6.56/1.24 rd(X, ld(mult(X, Z), X))
% 6.56/1.24 = { by lemma 6 }
% 6.56/1.24 mult(X, Z)
% 6.56/1.24
% 6.56/1.24 Lemma 28: mult(rd(X, ld(Y, Y)), Y) = mult(rd(X, Y), mult(Y, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(rd(X, ld(Y, Y)), Y)
% 6.56/1.24 = { by lemma 27 R->L }
% 6.56/1.24 rd(rd(X, ld(Y, Y)), ld(X, rd(X, Y)))
% 6.56/1.24 = { by lemma 25 R->L }
% 6.56/1.24 rd(mult(X, ld(Y, Y)), ld(X, rd(X, Y)))
% 6.56/1.24 = { by lemma 11 R->L }
% 6.56/1.24 rd(mult(X, mult(ld(X, rd(X, Y)), Y)), ld(X, rd(X, Y)))
% 6.56/1.24 = { by lemma 22 R->L }
% 6.56/1.24 mult(mult(X, ld(X, rd(X, Y))), rd(Y, ld(X, rd(X, Y))))
% 6.56/1.24 = { by axiom 3 (f01) }
% 6.56/1.24 mult(rd(X, Y), rd(Y, ld(X, rd(X, Y))))
% 6.56/1.24 = { by lemma 19 R->L }
% 6.56/1.24 mult(rd(X, Y), rd(Y, ld(Z, rd(Z, Y))))
% 6.56/1.24 = { by lemma 27 }
% 6.56/1.24 mult(rd(X, Y), mult(Y, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 29: mult(X, rd(Y, Y)) = rd(X, ld(Y, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(X, rd(Y, Y))
% 6.56/1.24 = { by lemma 23 R->L }
% 6.56/1.24 rd(mult(rd(X, Y), mult(Y, Y)), Y)
% 6.56/1.24 = { by lemma 28 R->L }
% 6.56/1.24 rd(mult(rd(X, ld(Y, Y)), Y), Y)
% 6.56/1.24 = { by axiom 2 (f04) }
% 6.56/1.24 rd(X, ld(Y, Y))
% 6.56/1.24
% 6.56/1.24 Lemma 30: rd(X, X) = ld(X, X).
% 6.56/1.24 Proof:
% 6.56/1.24 rd(X, X)
% 6.56/1.24 = { by axiom 1 (f02) R->L }
% 6.56/1.24 ld(Y, mult(Y, rd(X, X)))
% 6.56/1.24 = { by lemma 29 }
% 6.56/1.24 ld(Y, rd(Y, ld(X, X)))
% 6.56/1.24 = { by lemma 20 }
% 6.56/1.24 ld(X, X)
% 6.56/1.24
% 6.56/1.24 Lemma 31: rd(mult(X, X), ld(X, X)) = mult(X, X).
% 6.56/1.24 Proof:
% 6.56/1.24 rd(mult(X, X), ld(X, X))
% 6.56/1.24 = { by lemma 26 R->L }
% 6.56/1.24 mult(X, mult(ld(X, X), X))
% 6.56/1.24 = { by lemma 9 }
% 6.56/1.24 ld(rd(X, X), mult(X, X))
% 6.56/1.24 = { by lemma 30 }
% 6.56/1.24 ld(ld(X, X), mult(X, X))
% 6.56/1.24 = { by lemma 6 R->L }
% 6.56/1.24 ld(ld(X, X), mult(rd(X, ld(X, X)), X))
% 6.56/1.24 = { by lemma 28 }
% 6.56/1.24 ld(ld(X, X), mult(rd(X, X), mult(X, X)))
% 6.56/1.24 = { by lemma 30 }
% 6.56/1.24 ld(ld(X, X), mult(ld(X, X), mult(X, X)))
% 6.56/1.24 = { by axiom 1 (f02) }
% 6.56/1.24 mult(X, X)
% 6.56/1.24
% 6.56/1.24 Lemma 32: mult(rd(X, mult(Y, Z)), Y) = rd(rd(X, Y), rd(Z, Y)).
% 6.56/1.24 Proof:
% 6.56/1.24 mult(rd(X, mult(Y, Z)), Y)
% 6.56/1.24 = { by axiom 4 (f03) R->L }
% 6.56/1.24 mult(rd(X, mult(Y, mult(rd(Z, Y), Y))), Y)
% 6.56/1.24 = { by axiom 2 (f04) R->L }
% 6.56/1.25 rd(mult(mult(rd(X, mult(Y, mult(rd(Z, Y), Y))), Y), rd(Z, Y)), rd(Z, Y))
% 6.56/1.25 = { by lemma 21 R->L }
% 6.56/1.25 rd(rd(mult(rd(X, mult(Y, mult(rd(Z, Y), Y))), mult(Y, mult(rd(Z, Y), Y))), Y), rd(Z, Y))
% 6.56/1.25 = { by axiom 4 (f03) }
% 6.56/1.25 rd(rd(X, Y), rd(Z, Y))
% 6.56/1.25
% 6.56/1.25 Lemma 33: ld(ld(X, X), ld(X, X)) = ld(X, X).
% 6.56/1.25 Proof:
% 6.56/1.25 ld(ld(X, X), ld(X, X))
% 6.56/1.25 = { by lemma 12 R->L }
% 6.56/1.25 mult(ld(X, X), ld(X, X))
% 6.56/1.25 = { by lemma 11 R->L }
% 6.56/1.25 mult(ld(X, X), mult(ld(Y, rd(Y, X)), X))
% 6.56/1.25 = { by lemma 6 R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, ld(ld(Y, rd(Y, X)), X)), X))
% 6.56/1.25 = { by lemma 16 R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, ld(ld(mult(X, X), X), X)), X))
% 6.56/1.25 = { by axiom 1 (f02) R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, ld(ld(mult(X, X), X), ld(mult(X, X), mult(mult(X, X), X)))), X))
% 6.56/1.25 = { by lemma 6 R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, ld(ld(mult(X, X), X), ld(rd(X, ld(mult(X, X), X)), mult(mult(X, X), X)))), X))
% 6.56/1.25 = { by axiom 4 (f03) R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, ld(ld(mult(X, X), X), ld(rd(X, ld(mult(X, X), X)), mult(rd(mult(mult(X, X), X), ld(mult(X, X), X)), ld(mult(X, X), X))))), X))
% 6.56/1.25 = { by lemma 10 }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(ld(X, rd(mult(mult(X, X), X), ld(mult(X, X), X))), ld(mult(X, X), X))), X))
% 6.56/1.25 = { by lemma 26 R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(ld(X, mult(X, mult(ld(X, mult(X, X)), mult(X, X)))), ld(mult(X, X), X))), X))
% 6.56/1.25 = { by axiom 1 (f02) }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(mult(ld(X, mult(X, X)), mult(X, X)), ld(mult(X, X), X))), X))
% 6.56/1.25 = { by lemma 25 }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, rd(mult(ld(X, mult(X, X)), mult(X, X)), ld(X, mult(X, X)))), X))
% 6.56/1.25 = { by axiom 1 (f02) }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, rd(mult(X, mult(X, X)), ld(X, mult(X, X)))), X))
% 6.56/1.25 = { by lemma 26 R->L }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(mult(X, X), mult(ld(mult(X, X), X), X))), X))
% 6.56/1.25 = { by lemma 14 }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(mult(X, X), ld(X, X))), X))
% 6.56/1.25 = { by lemma 18 }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(X, mult(ld(X, X), X))), X))
% 6.56/1.25 = { by lemma 26 }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, rd(mult(X, X), ld(X, X))), X))
% 6.56/1.25 = { by lemma 31 }
% 6.56/1.25 mult(ld(X, X), mult(rd(X, mult(X, X)), X))
% 6.56/1.25 = { by lemma 32 }
% 6.56/1.25 mult(ld(X, X), rd(rd(X, X), rd(X, X)))
% 6.56/1.25 = { by lemma 30 }
% 6.56/1.25 mult(ld(X, X), ld(rd(X, X), rd(X, X)))
% 6.56/1.25 = { by lemma 30 }
% 6.56/1.25 mult(ld(X, X), ld(rd(X, X), ld(X, X)))
% 6.56/1.25 = { by lemma 30 }
% 6.56/1.25 mult(ld(X, X), ld(ld(X, X), ld(X, X)))
% 6.56/1.25 = { by axiom 3 (f01) }
% 6.56/1.25 ld(X, X)
% 6.56/1.25
% 6.56/1.25 Lemma 34: rd(ld(X, Y), ld(X, Y)) = ld(ld(Y, X), ld(Y, X)).
% 6.56/1.25 Proof:
% 6.56/1.25 rd(ld(X, Y), ld(X, Y))
% 6.56/1.25 = { by lemma 20 R->L }
% 6.56/1.25 rd(ld(Z, rd(Z, ld(Y, X))), ld(X, Y))
% 6.56/1.25 = { by lemma 25 R->L }
% 6.56/1.25 mult(ld(Z, rd(Z, ld(Y, X))), ld(Y, X))
% 6.56/1.25 = { by lemma 11 }
% 6.56/1.25 ld(ld(Y, X), ld(Y, X))
% 6.56/1.25
% 6.56/1.25 Lemma 35: ld(ld(X, Y), ld(X, Y)) = ld(ld(Y, X), ld(Y, X)).
% 6.56/1.25 Proof:
% 6.56/1.25 ld(ld(X, Y), ld(X, Y))
% 6.56/1.25 = { by lemma 34 R->L }
% 6.56/1.25 rd(ld(Y, X), ld(Y, X))
% 6.56/1.25 = { by lemma 30 }
% 6.56/1.25 ld(ld(Y, X), ld(Y, X))
% 6.56/1.25
% 6.56/1.25 Lemma 36: rd(rd(X, Y), rd(ld(Y, Z), Y)) = mult(rd(X, Z), Y).
% 6.56/1.25 Proof:
% 6.56/1.25 rd(rd(X, Y), rd(ld(Y, Z), Y))
% 6.56/1.25 = { by lemma 32 R->L }
% 6.56/1.25 mult(rd(X, mult(Y, ld(Y, Z))), Y)
% 6.56/1.25 = { by axiom 3 (f01) }
% 6.56/1.25 mult(rd(X, Z), Y)
% 6.56/1.25
% 6.56/1.25 Lemma 37: ld(mult(X, Y), rd(mult(X, Z), Y)) = rd(ld(Y, Z), Y).
% 6.56/1.25 Proof:
% 6.56/1.25 ld(mult(X, Y), rd(mult(X, Z), Y))
% 6.56/1.25 = { by axiom 3 (f01) R->L }
% 6.56/1.25 ld(mult(X, Y), rd(mult(X, mult(Y, ld(Y, Z))), Y))
% 6.56/1.25 = { by lemma 22 R->L }
% 6.56/1.25 ld(mult(X, Y), mult(mult(X, Y), rd(ld(Y, Z), Y)))
% 6.56/1.25 = { by axiom 1 (f02) }
% 6.56/1.25 rd(ld(Y, Z), Y)
% 6.56/1.25
% 6.56/1.25 Lemma 38: ld(mult(rd(X, Y), Z), rd(X, Z)) = rd(ld(Z, Y), Z).
% 6.56/1.25 Proof:
% 6.56/1.25 ld(mult(rd(X, Y), Z), rd(X, Z))
% 6.56/1.25 = { by axiom 4 (f03) R->L }
% 6.56/1.25 ld(mult(rd(X, Y), Z), rd(mult(rd(X, Y), Y), Z))
% 6.56/1.25 = { by lemma 37 }
% 6.56/1.25 rd(ld(Z, Y), Z)
% 6.56/1.25
% 6.56/1.25 Lemma 39: mult(ld(X, Y), mult(Z, ld(X, Y))) = ld(X, mult(mult(Y, Z), ld(X, Y))).
% 6.56/1.25 Proof:
% 6.56/1.25 mult(ld(X, Y), mult(Z, ld(X, Y)))
% 6.56/1.25 = { by axiom 1 (f02) R->L }
% 6.56/1.25 ld(X, mult(X, mult(ld(X, Y), mult(Z, ld(X, Y)))))
% 6.56/1.25 = { by lemma 17 }
% 6.56/1.25 ld(X, mult(mult(Y, Z), ld(X, Y)))
% 6.56/1.25
% 6.56/1.25 Lemma 40: ld(rd(X, Y), mult(rd(X, ld(Z, W)), Y)) = ld(rd(W, Y), mult(Z, Y)).
% 6.56/1.25 Proof:
% 6.56/1.25 ld(rd(X, Y), mult(rd(X, ld(Z, W)), Y))
% 6.56/1.25 = { by lemma 27 R->L }
% 6.56/1.25 ld(rd(X, Y), rd(rd(X, ld(Z, W)), ld(X, rd(X, Y))))
% 6.56/1.25 = { by lemma 25 R->L }
% 6.56/1.25 ld(rd(X, Y), rd(mult(X, ld(W, Z)), ld(X, rd(X, Y))))
% 6.56/1.25 = { by lemma 25 R->L }
% 6.56/1.25 ld(rd(X, Y), mult(mult(X, ld(W, Z)), ld(rd(X, Y), X)))
% 6.56/1.25 = { by lemma 39 R->L }
% 6.56/1.25 mult(ld(rd(X, Y), X), mult(ld(W, Z), ld(rd(X, Y), X)))
% 6.56/1.25 = { by lemma 9 }
% 6.56/1.25 ld(rd(W, ld(rd(X, Y), X)), mult(Z, ld(rd(X, Y), X)))
% 6.56/1.25 = { by lemma 25 }
% 6.56/1.25 ld(rd(W, ld(rd(X, Y), X)), rd(Z, ld(X, rd(X, Y))))
% 6.56/1.25 = { by lemma 13 }
% 6.56/1.25 ld(rd(W, Y), rd(Z, ld(X, rd(X, Y))))
% 6.56/1.25 = { by lemma 19 R->L }
% 6.56/1.25 ld(rd(W, Y), rd(Z, ld(V, rd(V, Y))))
% 6.56/1.25 = { by lemma 27 }
% 6.56/1.25 ld(rd(W, Y), mult(Z, Y))
% 6.56/1.25
% 6.56/1.25 Lemma 41: rd(rd(X, ld(Y, Y)), ld(Y, Y)) = X.
% 6.56/1.25 Proof:
% 6.56/1.25 rd(rd(X, ld(Y, Y)), ld(Y, Y))
% 6.56/1.25 = { by lemma 30 R->L }
% 6.56/1.25 rd(rd(X, ld(Y, Y)), rd(Y, Y))
% 6.56/1.25 = { by lemma 29 R->L }
% 6.56/1.25 rd(mult(X, rd(Y, Y)), rd(Y, Y))
% 6.56/1.25 = { by axiom 2 (f04) }
% 6.56/1.25 X
% 6.56/1.25
% 6.56/1.25 Lemma 42: rd(ld(X, ld(Y, Y)), ld(Y, Y)) = ld(Z, rd(Z, X)).
% 6.56/1.25 Proof:
% 6.56/1.25 rd(ld(X, ld(Y, Y)), ld(Y, Y))
% 6.56/1.25 = { by axiom 2 (f04) R->L }
% 6.56/1.25 rd(mult(rd(ld(X, ld(Y, Y)), ld(Y, Y)), X), X)
% 6.56/1.25 = { by lemma 36 R->L }
% 6.56/1.25 rd(rd(rd(ld(X, ld(Y, Y)), X), rd(ld(X, ld(Y, Y)), X)), X)
% 6.56/1.25 = { by lemma 30 }
% 6.56/1.25 rd(ld(rd(ld(X, ld(Y, Y)), X), rd(ld(X, ld(Y, Y)), X)), X)
% 6.56/1.25 = { by lemma 38 R->L }
% 6.56/1.25 rd(ld(rd(ld(X, ld(Y, Y)), X), ld(mult(rd(W, ld(Y, Y)), X), rd(W, X))), X)
% 6.56/1.25 = { by lemma 38 R->L }
% 6.56/1.25 rd(ld(ld(mult(rd(W, ld(Y, Y)), X), rd(W, X)), ld(mult(rd(W, ld(Y, Y)), X), rd(W, X))), X)
% 6.56/1.25 = { by lemma 12 R->L }
% 6.56/1.25 rd(mult(ld(rd(W, X), mult(rd(W, ld(Y, Y)), X)), ld(mult(rd(W, ld(Y, Y)), X), rd(W, X))), X)
% 6.56/1.25 = { by lemma 40 }
% 6.56/1.25 rd(mult(ld(rd(Y, X), mult(Y, X)), ld(mult(rd(W, ld(Y, Y)), X), rd(W, X))), X)
% 6.56/1.25 = { by lemma 25 }
% 6.56/1.25 rd(rd(ld(rd(Y, X), mult(Y, X)), ld(rd(W, X), mult(rd(W, ld(Y, Y)), X))), X)
% 6.56/1.25 = { by lemma 40 }
% 6.56/1.25 rd(rd(ld(rd(Y, X), mult(Y, X)), ld(rd(Y, X), mult(Y, X))), X)
% 6.56/1.25 = { by lemma 34 }
% 6.56/1.25 rd(ld(ld(mult(Y, X), rd(Y, X)), ld(mult(Y, X), rd(Y, X))), X)
% 6.56/1.25 = { by lemma 35 }
% 6.56/1.25 rd(ld(ld(rd(Y, X), mult(Y, X)), ld(rd(Y, X), mult(Y, X))), X)
% 6.56/1.25 = { by lemma 9 R->L }
% 6.56/1.25 rd(ld(ld(rd(Y, X), mult(Y, X)), mult(X, mult(ld(Y, Y), X))), X)
% 6.56/1.25 = { by lemma 41 R->L }
% 6.56/1.25 rd(ld(ld(rd(Y, X), mult(Y, X)), mult(rd(rd(X, ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), X))), X)
% 6.56/1.25 = { by axiom 4 (f03) R->L }
% 6.56/1.25 rd(ld(ld(rd(Y, X), mult(Y, X)), mult(rd(rd(X, ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), mult(rd(X, ld(Y, Y)), ld(Y, Y))))), X)
% 6.56/1.25 = { by lemma 7 }
% 6.56/1.25 rd(ld(ld(rd(Y, X), mult(Y, X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 20 R->L }
% 6.56/1.25 rd(ld(ld(X, rd(X, ld(mult(Y, X), rd(Y, X)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 3 (f01) R->L }
% 6.56/1.25 rd(ld(ld(X, rd(X, ld(mult(Y, X), rd(mult(Y, ld(Y, Y)), X)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 37 }
% 6.56/1.25 rd(ld(ld(X, rd(X, rd(ld(X, ld(Y, Y)), X))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 30 R->L }
% 6.56/1.25 rd(ld(ld(X, rd(X, rd(ld(X, rd(Y, Y)), X))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 2 (f04) R->L }
% 6.56/1.25 rd(ld(ld(X, rd(rd(mult(X, X), X), rd(ld(X, rd(Y, Y)), X))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 36 }
% 6.56/1.25 rd(ld(ld(X, mult(rd(mult(X, X), rd(Y, Y)), X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 27 R->L }
% 6.56/1.25 rd(ld(ld(X, rd(rd(mult(X, X), rd(Y, Y)), ld(V, rd(V, X)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 16 R->L }
% 6.56/1.25 rd(ld(ld(X, rd(rd(mult(X, X), rd(Y, Y)), ld(mult(X, X), X))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 1 (f02) R->L }
% 6.56/1.25 rd(ld(ld(X, rd(rd(mult(X, X), ld(X, mult(X, rd(Y, Y)))), ld(mult(X, X), X))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 3 (f01) R->L }
% 6.56/1.25 rd(ld(ld(X, rd(rd(mult(X, X), ld(X, mult(X, rd(Y, Y)))), ld(mult(X, X), mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X))))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 25 R->L }
% 6.56/1.25 rd(ld(ld(X, rd(mult(mult(X, X), ld(mult(X, rd(Y, Y)), X)), ld(mult(X, X), mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X))))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 3 (f01) R->L }
% 6.56/1.25 rd(ld(ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), rd(mult(mult(X, X), ld(mult(X, rd(Y, Y)), X)), ld(mult(X, X), mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X))))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 25 R->L }
% 6.56/1.25 rd(ld(ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(mult(mult(X, X), ld(mult(X, rd(Y, Y)), X)), ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(X, X)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 39 R->L }
% 6.56/1.25 rd(ld(mult(ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(X, X)), mult(ld(mult(X, rd(Y, Y)), X), ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(X, X)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 8 R->L }
% 6.56/1.25 rd(ld(ld(rd(mult(X, rd(Y, Y)), ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(X, X))), mult(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(X, X)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 3 (f01) }
% 6.56/1.25 rd(ld(ld(rd(mult(X, rd(Y, Y)), ld(mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), mult(X, X))), mult(X, X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 3 (f01) }
% 6.56/1.25 rd(ld(ld(rd(mult(X, rd(Y, Y)), ld(X, mult(X, X))), mult(X, X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 1 (f02) }
% 6.56/1.25 rd(ld(ld(rd(mult(X, rd(Y, Y)), X), mult(X, X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 4 (f03) R->L }
% 6.56/1.25 rd(ld(mult(rd(ld(rd(mult(X, rd(Y, Y)), X), mult(X, X)), ld(mult(X, rd(Y, Y)), X)), ld(mult(X, rd(Y, Y)), X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 9 R->L }
% 6.56/1.25 rd(ld(mult(rd(mult(X, mult(ld(mult(X, rd(Y, Y)), X), X)), ld(mult(X, rd(Y, Y)), X)), ld(mult(X, rd(Y, Y)), X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by axiom 3 (f01) R->L }
% 6.56/1.25 rd(ld(mult(rd(mult(X, mult(ld(mult(X, rd(Y, Y)), X), mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)))), ld(mult(X, rd(Y, Y)), X)), ld(mult(X, rd(Y, Y)), X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 21 }
% 6.56/1.25 rd(ld(mult(mult(mult(X, ld(mult(X, rd(Y, Y)), X)), mult(X, rd(Y, Y))), ld(mult(X, rd(Y, Y)), X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 25 }
% 6.56/1.25 rd(ld(mult(mult(rd(X, ld(X, mult(X, rd(Y, Y)))), mult(X, rd(Y, Y))), ld(mult(X, rd(Y, Y)), X)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.25 = { by lemma 25 }
% 6.56/1.25 rd(ld(rd(mult(rd(X, ld(X, mult(X, rd(Y, Y)))), mult(X, rd(Y, Y))), ld(X, mult(X, rd(Y, Y)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by axiom 1 (f02) }
% 6.56/1.26 rd(ld(rd(mult(rd(X, rd(Y, Y)), mult(X, rd(Y, Y))), ld(X, mult(X, rd(Y, Y)))), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by axiom 1 (f02) }
% 6.56/1.26 rd(ld(rd(mult(rd(X, rd(Y, Y)), mult(X, rd(Y, Y))), rd(Y, Y)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 29 }
% 6.56/1.26 rd(ld(rd(mult(rd(X, rd(Y, Y)), rd(X, ld(Y, Y))), rd(Y, Y)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 30 }
% 6.56/1.26 rd(ld(rd(mult(rd(X, rd(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 30 }
% 6.56/1.26 rd(ld(rd(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 31 R->L }
% 6.56/1.26 rd(ld(rd(rd(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y)))), ld(Y, Y)), mult(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by axiom 4 (f03) R->L }
% 6.56/1.26 rd(ld(rd(rd(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y)))), ld(Y, Y)), mult(mult(rd(mult(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y)))), ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y)))), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 8 }
% 6.56/1.26 rd(mult(ld(Y, Y), mult(ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 9 }
% 6.56/1.26 rd(ld(rd(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), X)
% 6.56/1.26 = { by lemma 41 }
% 6.56/1.26 rd(ld(X, mult(rd(X, ld(Y, Y)), ld(Y, Y))), X)
% 6.56/1.26 = { by axiom 4 (f03) }
% 6.56/1.26 rd(ld(X, X), X)
% 6.56/1.26 = { by lemma 15 }
% 6.56/1.26 ld(Z, rd(Z, X))
% 6.56/1.26
% 6.56/1.26 Lemma 43: ld(ld(X, ld(Y, Y)), ld(X, ld(Y, Y))) = ld(X, X).
% 6.56/1.26 Proof:
% 6.56/1.26 ld(ld(X, ld(Y, Y)), ld(X, ld(Y, Y)))
% 6.56/1.26 = { by lemma 35 R->L }
% 6.56/1.26 ld(ld(ld(Y, Y), X), ld(ld(Y, Y), X))
% 6.56/1.26 = { by lemma 30 R->L }
% 6.56/1.26 rd(ld(ld(Y, Y), X), ld(ld(Y, Y), X))
% 6.56/1.26 = { by lemma 24 R->L }
% 6.56/1.26 mult(ld(ld(Y, Y), X), ld(Z, rd(Z, ld(ld(Y, Y), X))))
% 6.56/1.26 = { by lemma 42 R->L }
% 6.56/1.26 mult(ld(ld(Y, Y), X), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y)))
% 6.56/1.26 = { by lemma 6 R->L }
% 6.56/1.26 mult(rd(ld(Y, Y), ld(ld(ld(Y, Y), X), ld(Y, Y))), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y)))
% 6.56/1.26 = { by axiom 4 (f03) R->L }
% 6.56/1.26 mult(rd(ld(Y, Y), mult(rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y)))
% 6.56/1.26 = { by lemma 32 }
% 6.56/1.26 rd(rd(ld(Y, Y), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y))), rd(ld(Y, Y), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y))))
% 6.56/1.26 = { by lemma 30 }
% 6.56/1.26 ld(rd(ld(Y, Y), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y))), rd(ld(Y, Y), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y))))
% 6.56/1.26 = { by lemma 42 }
% 6.56/1.26 ld(rd(ld(Y, Y), ld(W, rd(W, ld(ld(Y, Y), X)))), rd(ld(Y, Y), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y))))
% 6.56/1.26 = { by lemma 27 }
% 6.56/1.26 ld(mult(ld(Y, Y), ld(ld(Y, Y), X)), rd(ld(Y, Y), rd(ld(ld(ld(Y, Y), X), ld(Y, Y)), ld(Y, Y))))
% 6.56/1.26 = { by lemma 42 }
% 6.56/1.26 ld(mult(ld(Y, Y), ld(ld(Y, Y), X)), rd(ld(Y, Y), ld(V, rd(V, ld(ld(Y, Y), X)))))
% 6.56/1.26 = { by lemma 27 }
% 6.56/1.26 ld(mult(ld(Y, Y), ld(ld(Y, Y), X)), mult(ld(Y, Y), ld(ld(Y, Y), X)))
% 6.56/1.26 = { by axiom 3 (f01) }
% 6.56/1.26 ld(X, mult(ld(Y, Y), ld(ld(Y, Y), X)))
% 6.56/1.26 = { by axiom 3 (f01) }
% 6.56/1.26 ld(X, X)
% 6.56/1.26
% 6.56/1.26 Lemma 44: ld(Y, Y) = ld(X, X).
% 6.56/1.26 Proof:
% 6.56/1.26 ld(Y, Y)
% 6.56/1.26 = { by lemma 33 R->L }
% 6.56/1.26 ld(ld(Y, Y), ld(Y, Y))
% 6.56/1.26 = { by lemma 43 R->L }
% 6.56/1.26 ld(ld(ld(Y, Y), ld(ld(X, X), ld(X, X))), ld(ld(Y, Y), ld(ld(X, X), ld(X, X))))
% 6.56/1.26 = { by lemma 35 R->L }
% 6.56/1.26 ld(ld(ld(ld(X, X), ld(X, X)), ld(Y, Y)), ld(ld(ld(X, X), ld(X, X)), ld(Y, Y)))
% 6.56/1.26 = { by lemma 43 }
% 6.56/1.26 ld(ld(ld(X, X), ld(X, X)), ld(ld(X, X), ld(X, X)))
% 6.56/1.26 = { by lemma 33 }
% 6.56/1.26 ld(ld(X, X), ld(X, X))
% 6.56/1.26 = { by lemma 33 }
% 6.56/1.26 ld(X, X)
% 6.56/1.26
% 6.56/1.26 Goal 1 (goals): tuple(mult(X, x1(X)), mult(x1_2(X), X)) = tuple(x1(X), x1_2(X)).
% 6.56/1.26 The goal is true when:
% 6.56/1.26 X = ld(X, X)
% 6.56/1.26
% 6.56/1.26 Proof:
% 6.56/1.26 tuple(mult(ld(X, X), x1(ld(X, X))), mult(x1_2(ld(X, X)), ld(X, X)))
% 6.56/1.26 = { by lemma 44 }
% 6.56/1.26 tuple(mult(ld(x1(ld(X, X)), x1(ld(X, X))), x1(ld(X, X))), mult(x1_2(ld(X, X)), ld(X, X)))
% 6.56/1.26 = { by lemma 30 R->L }
% 6.56/1.26 tuple(mult(rd(x1(ld(X, X)), x1(ld(X, X))), x1(ld(X, X))), mult(x1_2(ld(X, X)), ld(X, X)))
% 6.56/1.26 = { by axiom 4 (f03) }
% 6.56/1.26 tuple(x1(ld(X, X)), mult(x1_2(ld(X, X)), ld(X, X)))
% 6.56/1.26 = { by lemma 44 }
% 6.56/1.26 tuple(x1(ld(X, X)), mult(x1_2(ld(X, X)), ld(x1_2(ld(X, X)), x1_2(ld(X, X)))))
% 6.56/1.26 = { by axiom 3 (f01) }
% 6.56/1.26 tuple(x1(ld(X, X)), x1_2(ld(X, X)))
% 6.56/1.26 % SZS output end Proof
% 6.56/1.26
% 6.56/1.26 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------