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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 COMPOS_1:141;
A7: IC (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))) = IC ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) by A6, FUNCT_4:14
.= 0 by COMPOS_1:142 ;
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 COMPOS_1:38;
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;
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 EXTPRO_1:4
.= Following ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))) by EXTPRO_1:3
.= Exec ((a =0_goto 4),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))) by A11 ;
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)
.= 4 by A2, A12, A8, SCMFSA_2:96 ;
A15: S₁[ 0 ]

proof

assume 0 <= LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 COMPOS_1:141;
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 EXTPRO_1:3
.= IC (I +* (Start-At (0,SCM+FSA))) by A16, FUNCT_4:14
.= 0 by COMPOS_1:142 ;
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, EXTPRO_1:3; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A15, A3);
then A17: S₁[ LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))))) by AMI_1:123;
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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))) by EXTPRO_1:4
.= 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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))))) by A19, T ;
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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))) + 1)))) by COMPOS_1:38;
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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))) + 1))) . (IC SCM+FSA)
.= (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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))) + 1)))) = goto 0 by A22, Y;
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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))) + 1))) by AMI_1:123;
Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(((1 + (LifeSpan ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))) + 1)))) by EXTPRO_1:4
.= 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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA))))))) + 1)))) by A23, T ;
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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(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 ((ProgramPart (s +* (I +* (Start-At (0,SCM+FSA))))),(s +* (I +* (Start-At (0,SCM+FSA)))))))) by A17, A21, SCMFSA6A:38 ;
:: thesis: verum