let s2 be State of SCM+FSA; :: thesis: not ProgramPart (s2 +* ((IC s2),(goto (IC s2)))) halts_on s2 +* ((IC s2),(goto (IC s2)))
set s1 = s2 +* ((IC s2),(goto (IC s2)));
defpred S₁[ Nat] means IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),$1)) = IC (s2 +* ((IC s2),(goto (IC s2))));
IC SCM+FSA <> IC s2 by COMPOS_1:3;
then A1: IC (s2 +* ((IC s2),(goto (IC s2)))) = IC s2 by FUNCT_7:34;
A2: IC s2 in dom s2 by COMPOS_1:22;

A3: now

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
Y: (ProgramPart (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) /. (IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) . (IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) by COMPOS_1:38;
T: ProgramPart (s2 +* ((IC s2),(goto (IC s2)))) = ProgramPart (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) by AMI_1:123;
assume S₁[n] ; :: thesis: S₁[n + 1]
then A4: CurInstr ((ProgramPart (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = (s2 +* ((IC s2),(goto (IC s2)))) . (IC (s2 +* ((IC s2),(goto (IC s2))))) by Y, AMI_1:54
.= goto (IC s2) by A2, A1, FUNCT_7:33 ;
IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),(n + 1))) = IC (Following ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)))) by EXTPRO_1:4
.= IC (s2 +* ((IC s2),(goto (IC s2)))) by A1, A4, T, SCMFSA_2:95 ;
hence S₁[n + 1] ; :: thesis: verum

end;

let n be Nat; :: according to EXTPRO_1:def 7 :: thesis: ( not IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) in proj1 (ProgramPart (s2 +* ((IC s2),(goto (IC s2))))) or not CurInstr ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = halt SCM+FSA )
A5: S₁[ 0 ] by EXTPRO_1:3;
assume IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) in dom (ProgramPart (s2 +* ((IC s2),(goto (IC s2))))) ; :: thesis: not CurInstr ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = halt SCM+FSA
reconsider n = n as Element of NAT by ORDINAL1:def 13;
ProgramPart (s2 +* ((IC s2),(goto (IC s2)))) = ProgramPart (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) by AMI_1:123;
then Y: (ProgramPart (s2 +* ((IC s2),(goto (IC s2))))) /. (IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) . (IC (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) by COMPOS_1:38;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A5, A3);
then CurInstr ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = (Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n)) . (IC (s2 +* ((IC s2),(goto (IC s2))))) by Y
.= (s2 +* ((IC s2),(goto (IC s2)))) . (IC (s2 +* ((IC s2),(goto (IC s2))))) by AMI_1:54
.= goto (IC s2) by A2, A1, FUNCT_7:33 ;
then CurInstr ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) <> halt SCM+FSA ;
hence not CurInstr ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(Comput ((ProgramPart (s2 +* ((IC s2),(goto (IC s2))))),(s2 +* ((IC s2),(goto (IC s2)))),n))) = halt SCM+FSA ; :: thesis: verum