let s be State of SCM+FSA ; :: thesis: for a being read-write Int-Location
for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s & s . a = 0 holds
DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))) + 3)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))
let a be read-write Int-Location ; :: thesis: for I being Program of SCM+FSA st I is_closed_on s & I is_halting_on s & s . a = 0 holds
DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))) + 3)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))
let I be Program of SCM+FSA ; :: thesis: ( I is_closed_on s & I is_halting_on s & s . a = 0 implies DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))) + 3)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) )
assume that
A1: ( I is_closed_on s & I is_halting_on s ) and
A2: s . a = 0 ; :: thesis: DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))) + 3)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))
set sI = s +* (I +* (Start-At 0 ,SCM+FSA ));
set s1 = s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ));
defpred S₁[ Nat] means ( $1 <= LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))) implies ( IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + $1)) = (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),$1)) + 4 & DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + $1)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),$1) ) );

A3: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: S₁[k] ; :: thesis: S₁[k + 1]

now

A5: k + 0 < k + 1 by XREAL_1:8;
assume k + 1 <= LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))) ; :: thesis: ( IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1))) + 4 & DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) )
then k < LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))) by A5, XXREAL_0:2;
hence ( IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1))) + 4 & DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + k) + 1)) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(k + 1)) ) by A1, A4, SCMFSA_9:19; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

set i = a =0_goto 4;
set s2 = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),1;
A6: IC SCM+FSA in dom ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
A7: IC (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . (IC SCM+FSA ) by AMI_1:def 15
.= ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by A6, FUNCT_4:14
.= 0 by SF_MASTR:66 ;
not a in dom ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) by SCMFSA6B:12;
then A8: (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . a = s . a by FUNCT_4:12;
set loc4 = (card I) + 4;
while=0 a,I c= (while=0 a,I) +* (Start-At 0 ,SCM+FSA ) by SCMFSA8A:9;
then A9: dom (while=0 a,I) c= dom ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) by GRFUNC_1:8;
Y: (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))) /. (IC (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))) = (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . (IC (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))) by AMI_1:150;
A10: 0 in dom (while=0 a,I) by SCMFSA_9:10;
then (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . 0 = ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) . 0 by A9, FUNCT_4:14
.= (while=0 a,I) . 0 by A10, SCMFSA6B:7
.= a =0_goto 4 by SCMFSA_9:11 ;
then A11: CurInstr (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = a =0_goto 4 by A7, Y, AMI_1:def 16;
A12: Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(0 + 1) = Following (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),0 ) by AMI_1:14
.= Following (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) by AMI_1:13
.= Exec (a =0_goto 4),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) by A11, AMI_1:def 18 ;
then ( ( for c being Int-Location holds (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) . c = (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . c ) & ( for f being FinSeq-Location holds (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) . f = (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . f ) ) by SCMFSA_2:96;
then A13: DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) = DataPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) by SCMFSA6A:38
.= DataPart s by SCMFSA8A:11
.= DataPart (s +* (I +* (Start-At 0 ,SCM+FSA ))) by SCMFSA8A:11 ;
A14: IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),1) . (IC SCM+FSA ) by AMI_1:def 15
.= 4 by A2, A12, A8, SCMFSA_2:96 ;
A15: S₁[ 0 ]

proof

assume 0 <= LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))) ; :: thesis: ( IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 )) + 4 & DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) )
A16: IC SCM+FSA in dom (I +* (Start-At 0 ,SCM+FSA )) by SF_MASTR:65;
IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) = IC (s +* (I +* (Start-At 0 ,SCM+FSA ))) by AMI_1:13
.= (s +* (I +* (Start-At 0 ,SCM+FSA ))) . (IC SCM+FSA ) by AMI_1:def 15
.= (I +* (Start-At 0 ,SCM+FSA )) . (IC SCM+FSA ) by A16, FUNCT_4:14
.= 0 by SF_MASTR:66 ;
hence ( IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = (IC (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 )) + 4 & DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + 0 )) = DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),0 ) ) by A14, A13, AMI_1:13; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A15, A3);
then A17: S₁[ LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))] ;
set s4 = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1);
set s3 = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1);
A18: (card I) + 4 in dom (while=0 a,I) by SCMFSA_9:13;
set s2 = Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))));
A19: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))) = goto ((card I) + 4) by A1, A17, SCMFSA_9:20;
T: ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))) by AMI_1:144;
A20: Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) = Following (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))) by AMI_1:14
.= Exec (goto ((card I) + 4)),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))) by A19, T, AMI_1:def 18 ;
then A21: ( ( for c being Int-Location holds (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . c = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))) . c ) & ( for f being FinSeq-Location holds (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . f = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))))) . f ) ) by SCMFSA_2:95;
A22: (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . ((card I) + 4) = (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) . ((card I) + 4) by AMI_1:54
.= ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )) . ((card I) + 4) by A9, A18, FUNCT_4:14
.= (while=0 a,I) . ((card I) + 4) by A18, SCMFSA6B:7
.= goto 0 by SCMFSA_9:21 ;
Y: (ProgramPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))) /. (IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))) = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . (IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))) by AMI_1:150;
IC (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . (IC SCM+FSA ) by AMI_1:def 15
.= (card I) + 4 by A20, SCMFSA_2:95 ;
then A23: CurInstr (ProgramPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) = goto 0 by A22, Y, AMI_1:def 16;
T: ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))) = ProgramPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) by AMI_1:144;
Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1) = Following (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) by AMI_1:14
.= Exec (goto 0 ),(Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) by A23, T, AMI_1:def 18 ;
then ( ( for c being Int-Location holds (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1)) . c = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . c ) & ( for f being FinSeq-Location holds (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),(((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1) + 1)) . f = (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) . f ) ) by SCMFSA_2:95;
hence DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA )))) + 3)) = DataPart (Comput (ProgramPart (s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA )))),(s +* ((while=0 a,I) +* (Start-At 0 ,SCM+FSA ))),((1 + (LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) + 1)) by SCMFSA6A:38
.= DataPart (Comput (ProgramPart (s +* (I +* (Start-At 0 ,SCM+FSA )))),(s +* (I +* (Start-At 0 ,SCM+FSA ))),(LifeSpan (s +* (I +* (Start-At 0 ,SCM+FSA ))))) by A17, A21, SCMFSA6A:38 ;
:: thesis: verum