TSTP Solution File: SEU303+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SEU303+2 : TPTP v8.1.2. Released v3.3.0.
% 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 : n032.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 17:52:06 EDT 2023
% Result : Theorem 175.60s 23.13s
% Output : Proof 176.09s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU303+2 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.30 % Computer : n032.cluster.edu
% 0.12/0.30 % Model : x86_64 x86_64
% 0.12/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.30 % Memory : 8042.1875MB
% 0.12/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.30 % CPULimit : 300
% 0.12/0.30 % WCLimit : 300
% 0.12/0.30 % DateTime : Wed Aug 23 15:38:11 EDT 2023
% 0.12/0.30 % CPUTime :
% 175.60/23.13 Command-line arguments: --flatten
% 175.60/23.13
% 175.60/23.13 % SZS status Theorem
% 175.60/23.13
% 176.09/23.16 % SZS output start Proof
% 176.09/23.16 Take the following subset of the input axioms:
% 176.09/23.16 fof(fc12_relat_1, axiom, empty(empty_set) & (relation(empty_set) & relation_empty_yielding(empty_set))).
% 176.09/23.16 fof(fc13_finset_1, axiom, ![B, A2]: ((relation(A2) & (function(A2) & finite(B))) => finite(relation_image(A2, B)))).
% 176.09/23.16 fof(fc1_ordinal2, axiom, epsilon_transitive(omega) & (epsilon_connected(omega) & (ordinal(omega) & ~empty(omega)))).
% 176.09/23.17 fof(rc1_arytm_3, axiom, ?[A]: (~empty(A) & (epsilon_transitive(A) & (epsilon_connected(A) & (ordinal(A) & natural(A)))))).
% 176.09/23.17 fof(rc1_ordinal1, axiom, ?[A3]: (epsilon_transitive(A3) & (epsilon_connected(A3) & ordinal(A3)))).
% 176.09/23.17 fof(rc3_relat_1, axiom, ?[A3]: (relation(A3) & relation_empty_yielding(A3))).
% 176.09/23.17 fof(t146_relat_1, lemma, ![A2_2]: (relation(A2_2) => relation_image(A2_2, relation_dom(A2_2))=relation_rng(A2_2))).
% 176.09/23.17 fof(t26_finset_1, conjecture, ![A3]: ((relation(A3) & function(A3)) => (finite(relation_dom(A3)) => finite(relation_rng(A3))))).
% 176.09/23.17
% 176.09/23.17 Now clausify the problem and encode Horn clauses using encoding 3 of
% 176.09/23.17 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 176.09/23.17 We repeatedly replace C & s=t => u=v by the two clauses:
% 176.09/23.17 fresh(y, y, x1...xn) = u
% 176.09/23.17 C => fresh(s, t, x1...xn) = v
% 176.09/23.17 where fresh is a fresh function symbol and x1..xn are the free
% 176.09/23.17 variables of u and v.
% 176.09/23.17 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 176.09/23.17 input problem has no model of domain size 1).
% 176.09/23.17
% 176.09/23.17 The encoding turns the above axioms into the following unit equations and goals:
% 176.09/23.17
% 176.09/23.17 Axiom 1 (rc3_relat_1_1): relation_empty_yielding(a4) = true2.
% 176.09/23.17 Axiom 2 (fc12_relat_1_2): relation_empty_yielding(empty_set) = true2.
% 176.09/23.17 Axiom 3 (rc1_arytm_3_3): natural(a18) = true2.
% 176.09/23.17 Axiom 4 (rc1_ordinal1_1): epsilon_transitive(a15) = true2.
% 176.09/23.17 Axiom 5 (rc1_arytm_3_1): epsilon_transitive(a18) = true2.
% 176.09/23.17 Axiom 6 (fc1_ordinal2_1): epsilon_transitive(omega) = true2.
% 176.09/23.17 Axiom 7 (rc1_ordinal1): ordinal(a15) = true2.
% 176.09/23.17 Axiom 8 (rc1_arytm_3): ordinal(a18) = true2.
% 176.09/23.17 Axiom 9 (fc1_ordinal2): ordinal(omega) = true2.
% 176.09/23.17 Axiom 10 (t26_finset_1_1): function(a) = true2.
% 176.09/23.17 Axiom 11 (t26_finset_1_2): relation(a) = true2.
% 176.09/23.17 Axiom 12 (t26_finset_1): finite(relation_dom(a)) = true2.
% 176.09/23.17 Axiom 13 (t146_relat_1): fresh251(X, X, Y) = relation_rng(Y).
% 176.09/23.17 Axiom 14 (fc13_finset_1): fresh1165(X, X, Y, Z) = true2.
% 176.09/23.17 Axiom 15 (fc13_finset_1): fresh531(X, X, Y, Z) = finite(relation_image(Y, Z)).
% 176.09/23.17 Axiom 16 (t146_relat_1): fresh251(relation(X), true2, X) = relation_image(X, relation_dom(X)).
% 176.09/23.17 Axiom 17 (fc13_finset_1): fresh1164(X, X, Y, Z) = fresh1165(finite(Z), true2, Y, Z).
% 176.09/23.17 Axiom 18 (fc13_finset_1): fresh1164(relation(X), true2, X, Y) = fresh531(function(X), true2, X, Y).
% 176.09/23.17
% 176.09/23.17 Lemma 19: relation_empty_yielding(empty_set) = relation_empty_yielding(a4).
% 176.09/23.17 Proof:
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by axiom 2 (fc12_relat_1_2) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17
% 176.09/23.17 Lemma 20: natural(a18) = relation_empty_yielding(empty_set).
% 176.09/23.17 Proof:
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by axiom 3 (rc1_arytm_3_3) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17
% 176.09/23.17 Lemma 21: epsilon_transitive(a15) = natural(a18).
% 176.09/23.17 Proof:
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17 = { by axiom 4 (rc1_ordinal1_1) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17
% 176.09/23.17 Lemma 22: epsilon_transitive(a18) = epsilon_transitive(a15).
% 176.09/23.17 Proof:
% 176.09/23.17 epsilon_transitive(a18)
% 176.09/23.17 = { by axiom 5 (rc1_arytm_3_1) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17
% 176.09/23.17 Lemma 23: epsilon_transitive(omega) = epsilon_transitive(a18).
% 176.09/23.17 Proof:
% 176.09/23.17 epsilon_transitive(omega)
% 176.09/23.17 = { by axiom 6 (fc1_ordinal2_1) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17 = { by lemma 22 R->L }
% 176.09/23.17 epsilon_transitive(a18)
% 176.09/23.17
% 176.09/23.17 Lemma 24: ordinal(a15) = epsilon_transitive(omega).
% 176.09/23.17 Proof:
% 176.09/23.17 ordinal(a15)
% 176.09/23.17 = { by axiom 7 (rc1_ordinal1) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17 = { by lemma 22 R->L }
% 176.09/23.17 epsilon_transitive(a18)
% 176.09/23.17 = { by lemma 23 R->L }
% 176.09/23.17 epsilon_transitive(omega)
% 176.09/23.17
% 176.09/23.17 Lemma 25: ordinal(a18) = ordinal(a15).
% 176.09/23.17 Proof:
% 176.09/23.17 ordinal(a18)
% 176.09/23.17 = { by axiom 8 (rc1_arytm_3) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17 = { by lemma 22 R->L }
% 176.09/23.17 epsilon_transitive(a18)
% 176.09/23.17 = { by lemma 23 R->L }
% 176.09/23.17 epsilon_transitive(omega)
% 176.09/23.17 = { by lemma 24 R->L }
% 176.09/23.17 ordinal(a15)
% 176.09/23.17
% 176.09/23.17 Lemma 26: ordinal(omega) = ordinal(a18).
% 176.09/23.17 Proof:
% 176.09/23.17 ordinal(omega)
% 176.09/23.17 = { by axiom 9 (fc1_ordinal2) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17 = { by lemma 22 R->L }
% 176.09/23.17 epsilon_transitive(a18)
% 176.09/23.17 = { by lemma 23 R->L }
% 176.09/23.17 epsilon_transitive(omega)
% 176.09/23.17 = { by lemma 24 R->L }
% 176.09/23.17 ordinal(a15)
% 176.09/23.17 = { by lemma 25 R->L }
% 176.09/23.17 ordinal(a18)
% 176.09/23.17
% 176.09/23.17 Lemma 27: relation(a) = ordinal(omega).
% 176.09/23.17 Proof:
% 176.09/23.17 relation(a)
% 176.09/23.17 = { by axiom 11 (t26_finset_1_2) }
% 176.09/23.17 true2
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 relation_empty_yielding(a4)
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 relation_empty_yielding(empty_set)
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 natural(a18)
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 epsilon_transitive(a15)
% 176.09/23.17 = { by lemma 22 R->L }
% 176.09/23.17 epsilon_transitive(a18)
% 176.09/23.17 = { by lemma 23 R->L }
% 176.09/23.17 epsilon_transitive(omega)
% 176.09/23.17 = { by lemma 24 R->L }
% 176.09/23.17 ordinal(a15)
% 176.09/23.17 = { by lemma 25 R->L }
% 176.09/23.17 ordinal(a18)
% 176.09/23.17 = { by lemma 26 R->L }
% 176.09/23.17 ordinal(omega)
% 176.09/23.17
% 176.09/23.17 Goal 1 (t26_finset_1_3): finite(relation_rng(a)) = true2.
% 176.09/23.17 Proof:
% 176.09/23.17 finite(relation_rng(a))
% 176.09/23.17 = { by axiom 13 (t146_relat_1) R->L }
% 176.09/23.17 finite(fresh251(ordinal(omega), ordinal(omega), a))
% 176.09/23.17 = { by lemma 27 R->L }
% 176.09/23.17 finite(fresh251(relation(a), ordinal(omega), a))
% 176.09/23.17 = { by lemma 26 }
% 176.09/23.17 finite(fresh251(relation(a), ordinal(a18), a))
% 176.09/23.17 = { by lemma 25 }
% 176.09/23.17 finite(fresh251(relation(a), ordinal(a15), a))
% 176.09/23.17 = { by lemma 24 }
% 176.09/23.17 finite(fresh251(relation(a), epsilon_transitive(omega), a))
% 176.09/23.17 = { by lemma 23 }
% 176.09/23.17 finite(fresh251(relation(a), epsilon_transitive(a18), a))
% 176.09/23.17 = { by lemma 22 }
% 176.09/23.17 finite(fresh251(relation(a), epsilon_transitive(a15), a))
% 176.09/23.17 = { by lemma 21 }
% 176.09/23.17 finite(fresh251(relation(a), natural(a18), a))
% 176.09/23.17 = { by lemma 20 }
% 176.09/23.17 finite(fresh251(relation(a), relation_empty_yielding(empty_set), a))
% 176.09/23.17 = { by lemma 19 }
% 176.09/23.17 finite(fresh251(relation(a), relation_empty_yielding(a4), a))
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) }
% 176.09/23.17 finite(fresh251(relation(a), true2, a))
% 176.09/23.17 = { by axiom 16 (t146_relat_1) }
% 176.09/23.17 finite(relation_image(a, relation_dom(a)))
% 176.09/23.17 = { by axiom 15 (fc13_finset_1) R->L }
% 176.09/23.17 fresh531(ordinal(omega), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 26 }
% 176.09/23.17 fresh531(ordinal(a18), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 25 }
% 176.09/23.17 fresh531(ordinal(a15), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 24 }
% 176.09/23.17 fresh531(epsilon_transitive(omega), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 23 }
% 176.09/23.17 fresh531(epsilon_transitive(a18), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 22 }
% 176.09/23.17 fresh531(epsilon_transitive(a15), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 21 }
% 176.09/23.17 fresh531(natural(a18), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 20 }
% 176.09/23.17 fresh531(relation_empty_yielding(empty_set), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 19 }
% 176.09/23.17 fresh531(relation_empty_yielding(a4), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) }
% 176.09/23.17 fresh531(true2, ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by axiom 10 (t26_finset_1_1) R->L }
% 176.09/23.17 fresh531(function(a), ordinal(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 26 }
% 176.09/23.17 fresh531(function(a), ordinal(a18), a, relation_dom(a))
% 176.09/23.17 = { by lemma 25 }
% 176.09/23.17 fresh531(function(a), ordinal(a15), a, relation_dom(a))
% 176.09/23.17 = { by lemma 24 }
% 176.09/23.17 fresh531(function(a), epsilon_transitive(omega), a, relation_dom(a))
% 176.09/23.17 = { by lemma 23 }
% 176.09/23.17 fresh531(function(a), epsilon_transitive(a18), a, relation_dom(a))
% 176.09/23.17 = { by lemma 22 }
% 176.09/23.17 fresh531(function(a), epsilon_transitive(a15), a, relation_dom(a))
% 176.09/23.17 = { by lemma 21 }
% 176.09/23.17 fresh531(function(a), natural(a18), a, relation_dom(a))
% 176.09/23.17 = { by lemma 20 }
% 176.09/23.17 fresh531(function(a), relation_empty_yielding(empty_set), a, relation_dom(a))
% 176.09/23.17 = { by lemma 19 }
% 176.09/23.17 fresh531(function(a), relation_empty_yielding(a4), a, relation_dom(a))
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) }
% 176.09/23.17 fresh531(function(a), true2, a, relation_dom(a))
% 176.09/23.17 = { by axiom 18 (fc13_finset_1) R->L }
% 176.09/23.17 fresh1164(relation(a), true2, a, relation_dom(a))
% 176.09/23.17 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.17 fresh1164(relation(a), relation_empty_yielding(a4), a, relation_dom(a))
% 176.09/23.17 = { by lemma 19 R->L }
% 176.09/23.17 fresh1164(relation(a), relation_empty_yielding(empty_set), a, relation_dom(a))
% 176.09/23.17 = { by lemma 20 R->L }
% 176.09/23.17 fresh1164(relation(a), natural(a18), a, relation_dom(a))
% 176.09/23.17 = { by lemma 21 R->L }
% 176.09/23.17 fresh1164(relation(a), epsilon_transitive(a15), a, relation_dom(a))
% 176.09/23.18 = { by lemma 22 R->L }
% 176.09/23.18 fresh1164(relation(a), epsilon_transitive(a18), a, relation_dom(a))
% 176.09/23.18 = { by lemma 23 R->L }
% 176.09/23.18 fresh1164(relation(a), epsilon_transitive(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 24 R->L }
% 176.09/23.18 fresh1164(relation(a), ordinal(a15), a, relation_dom(a))
% 176.09/23.18 = { by lemma 25 R->L }
% 176.09/23.18 fresh1164(relation(a), ordinal(a18), a, relation_dom(a))
% 176.09/23.18 = { by lemma 26 R->L }
% 176.09/23.18 fresh1164(relation(a), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 27 }
% 176.09/23.18 fresh1164(ordinal(omega), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by axiom 17 (fc13_finset_1) }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), true2, a, relation_dom(a))
% 176.09/23.18 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), relation_empty_yielding(a4), a, relation_dom(a))
% 176.09/23.18 = { by lemma 19 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), relation_empty_yielding(empty_set), a, relation_dom(a))
% 176.09/23.18 = { by lemma 20 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), natural(a18), a, relation_dom(a))
% 176.09/23.18 = { by lemma 21 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), epsilon_transitive(a15), a, relation_dom(a))
% 176.09/23.18 = { by lemma 22 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), epsilon_transitive(a18), a, relation_dom(a))
% 176.09/23.18 = { by lemma 23 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), epsilon_transitive(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 24 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), ordinal(a15), a, relation_dom(a))
% 176.09/23.18 = { by lemma 25 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), ordinal(a18), a, relation_dom(a))
% 176.09/23.18 = { by lemma 26 R->L }
% 176.09/23.18 fresh1165(finite(relation_dom(a)), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by axiom 12 (t26_finset_1) }
% 176.09/23.18 fresh1165(true2, ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by axiom 1 (rc3_relat_1_1) R->L }
% 176.09/23.18 fresh1165(relation_empty_yielding(a4), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 19 R->L }
% 176.09/23.18 fresh1165(relation_empty_yielding(empty_set), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 20 R->L }
% 176.09/23.18 fresh1165(natural(a18), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 21 R->L }
% 176.09/23.18 fresh1165(epsilon_transitive(a15), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 22 R->L }
% 176.09/23.18 fresh1165(epsilon_transitive(a18), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 23 R->L }
% 176.09/23.18 fresh1165(epsilon_transitive(omega), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 24 R->L }
% 176.09/23.18 fresh1165(ordinal(a15), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 25 R->L }
% 176.09/23.18 fresh1165(ordinal(a18), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by lemma 26 R->L }
% 176.09/23.18 fresh1165(ordinal(omega), ordinal(omega), a, relation_dom(a))
% 176.09/23.18 = { by axiom 14 (fc13_finset_1) }
% 176.09/23.18 true2
% 176.09/23.18 % SZS output end Proof
% 176.09/23.18
% 176.09/23.18 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------