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 (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = goto (insloc 0 ) & ( for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA ) )
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s & I is_halting_on s implies ( CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = goto (insloc 0 ) & ( for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA ) ) )
set s1 = s +* (I +* (Start-At (insloc 0 )));
set s2 = s +* ((loop I) +* (Start-At (insloc 0 )));
assume A1: ( I is_closed_on s & I is_halting_on s ) ; :: thesis: ( CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = goto (insloc 0 ) & ( for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA ) )
then A2: s +* (I +* (Start-At (insloc 0 ))) is halting by SCMFSA7B:def 8;
set k = LifeSpan (s +* (I +* (Start-At (insloc 0 ))));
A3: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = halt SCM+FSA by A2, AMI_1:def 46;
A4: not IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom (Start-At (insloc 0 )) by AMI_1:137;
A5: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom I by A1, SCMFSA7B:def 7;
then A6: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom (I +* (Start-At (insloc 0 ))) by FUNCT_4:13;
A7: CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = (s +* (I +* (Start-At (insloc 0 )))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by AMI_1:54
.= (I +* (Start-At (insloc 0 ))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A6, FUNCT_4:14
.= I . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A4, FUNCT_4:12 ;
dom (loop I) = dom I by FUNCT_4:105;
then A8: IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) in dom ((loop I) +* (Start-At (insloc 0 ))) by A5, FUNCT_4:13;
X: rng I c= the Instructions of SCM+FSA by AMI_1:118;
y: dom (id the Instructions of SCM+FSA ) = the Instructions of SCM+FSA by RELAT_1:71;
thus A11: CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) = (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 )))))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A1, Th109, AMI_1:121
.= (s +* ((loop I) +* (Start-At (insloc 0 )))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by AMI_1:54
.= ((loop I) +* (Start-At (insloc 0 ))) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A8, FUNCT_4:14
.= (loop I) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by A4, FUNCT_4:12
.= (((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto (insloc 0 ))) * I) . (IC (Computation (s +* (I +* (Start-At (insloc 0 )))),(LifeSpan (s +* (I +* (Start-At (insloc 0 ))))))) by X, FUNCT_7:118
.= ((id the Instructions of SCM+FSA ) +* (halt SCM+FSA ),(goto (insloc 0 ))) . (halt SCM+FSA ) by A3, A5, A7, FUNCT_1:23
.= goto (insloc 0 ) by y, FUNCT_7:33 ; :: thesis: for m being Element of NAT st m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) holds
CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA
let m be Element of NAT ; :: thesis: ( m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) implies CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA )
assume A12: m <= LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA

per cases ( m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) or m = LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ) by A12, XXREAL_0:1;

suppose A13: m < LifeSpan (s +* (I +* (Start-At (insloc 0 )))) ; :: thesis: CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA

then CurInstr (Computation (s +* (I +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA by A2, AMI_1:def 46;
hence CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA by A1, A13, Th110; :: thesis: verum

end;

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

hence CurInstr (Computation (s +* ((loop I) +* (Start-At (insloc 0 )))),m) <> halt SCM+FSA by A11, SCMFSA_2:47, SCMFSA_2:124; :: thesis: verum

end;

end;