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 AMI_1:def 46;
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, AMI_1:def 46;
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, SCMFSA_2:47, SCMFSA_2:124; :: thesis: verum

end;

end;