A1: dom (id the Instructions of SCM+FSA) = the Instructions of SCM+FSA by RELAT_1:71;
let s be State of SCM+FSA; :: thesis: for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s holds
( CurInstr ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA ) )
let I be Program of SCM+FSA; :: thesis: ( I is_closed_on s & I is_halting_on s implies ( CurInstr ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA ) ) )
set s1 = s +* (I +* (Start-At (0,SCM+FSA)));
set s2 = s +* ((loop I) +* (Start-At (0,SCM+FSA)));
assume that
A2: I is_closed_on s and
A3: I is_halting_on s ; :: thesis: ( CurInstr ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = goto 0 & ( for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA ) )
set k = LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))));
A4: not IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom (Start-At (0,SCM+FSA)) by COMPOS_1:29;
A5: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom I by A2, SCMFSA7B:def 7;
then A6: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom (I +* (Start-At (0,SCM+FSA))) by FUNCT_4:13;
dom (loop I) = dom I by FUNCT_4:105;
then A7: IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) in dom ((loop I) +* (Start-At (0,SCM+FSA))) by A5, FUNCT_4:13;
A8: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) halts_on s +* (I +* (Start-At (0,SCM+FSA))) by A3, SCMFSA7B:def 8;
then A9: CurInstr ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = halt SCM+FSA by EXTPRO_1:def 14;
Y: (ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) /. (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by COMPOS_1:38;
TX1: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) by AMI_1:123;
A10: CurInstr ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (s +* (I +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by Y, TX1, AMI_1:54
.= (I +* (Start-At (0,SCM+FSA))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by A6, FUNCT_4:14
.= I . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by A4, FUNCT_4:12 ;
A11: rng I c= the Instructions of SCM+FSA by RELAT_1:def 19;
Y: (ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) /. (IC (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by COMPOS_1:38;
TX2: ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) by AMI_1:123;
thus A12: CurInstr ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) = (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by A2, A3, Th109, Y, TX2, COMPOS_1:24
.= (s +* ((loop I) +* (Start-At (0,SCM+FSA)))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by AMI_1:54
.= ((loop I) +* (Start-At (0,SCM+FSA))) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by A7, FUNCT_4:14
.= (loop I) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by A4, FUNCT_4:12
.= (((id the Instructions of SCM+FSA) +* ((halt SCM+FSA),(goto 0))) * I) . (IC (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),(LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by A11, FUNCT_7:118
.= ((id the Instructions of SCM+FSA) +* ((halt SCM+FSA),(goto 0))) . (halt SCM+FSA) by A9, A5, A10, FUNCT_1:23
.= goto 0 by A1, FUNCT_7:33 ; :: thesis: for m being Element of NAT st m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) holds
CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA
let m be Element of NAT ; :: thesis: ( m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) implies CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA )
assume A13: m <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ; :: thesis: CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA
TX1: ProgramPart (s +* (I +* (Start-At (0,SCM+FSA)))) = ProgramPart (Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m)) by AMI_1:123;

per cases ( m < LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) or m = LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ) by A13, XXREAL_0:1;

suppose A14: m < LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ; :: thesis: CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA

then CurInstr ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(Comput ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA by A8, EXTPRO_1:def 14;
hence CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA by A2, A3, A14, Th110, TX1; :: thesis: verum

end;

suppose m = LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))) ; :: thesis: CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA

hence CurInstr ((ProgramPart (Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))),(Comput ((ProgramPart (s +* ((loop I) +* (Start-At (0,SCM+FSA))))),(s +* ((loop I) +* (Start-At (0,SCM+FSA)))),m))) <> halt SCM+FSA by A12, TX2; :: thesis: verum

end;

end;