set D = Int-Locations \/ FinSeq-Locations;
let I be Program of SCM+FSA; :: thesis: for a being read-write Int-Location
for s being State of SCM+FSA
for k, n being Element of NAT st IC ((StepWhile=0 (a,I,s)) . k) = 0 & (StepWhile=0 (a,I,s)) . k = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),n) holds
( (StepWhile=0 (a,I,s)) . k = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) & (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) )
let a be read-write Int-Location ; :: thesis: for s being State of SCM+FSA
for k, n being Element of NAT st IC ((StepWhile=0 (a,I,s)) . k) = 0 & (StepWhile=0 (a,I,s)) . k = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),n) holds
( (StepWhile=0 (a,I,s)) . k = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) & (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) )
let s be State of SCM+FSA; :: thesis: for k, n being Element of NAT st IC ((StepWhile=0 (a,I,s)) . k) = 0 & (StepWhile=0 (a,I,s)) . k = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),n) holds
( (StepWhile=0 (a,I,s)) . k = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) & (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) )
let k, n be Element of NAT ; :: thesis: ( IC ((StepWhile=0 (a,I,s)) . k) = 0 & (StepWhile=0 (a,I,s)) . k = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),n) implies ( (StepWhile=0 (a,I,s)) . k = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) & (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) ) )
set s1 = s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)));
set sk = (StepWhile=0 (a,I,s)) . k;
set s2 = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)));
assume A1: IC ((StepWhile=0 (a,I,s)) . k) = 0 ; :: thesis: ( not (StepWhile=0 (a,I,s)) . k = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),n) or ( (StepWhile=0 (a,I,s)) . k = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) & (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) ) )
A2: IC SCM+FSA in dom ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) by COMPOS_1:141;
assume A3: (StepWhile=0 (a,I,s)) . k = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),n) ; :: thesis: ( (StepWhile=0 (a,I,s)) . k = ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) & (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) )
then ProgramPart ((StepWhile=0 (a,I,s)) . k) = ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))) by AMI_1:123;
then A4: ( DataPart (((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))) = DataPart ((StepWhile=0 (a,I,s)) . k) & ProgramPart (((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))) = ProgramPart ((StepWhile=0 (a,I,s)) . k) ) by FUNCT_4:100, SCMFSA8A:11;
T: ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))) = ProgramPart ((StepWhile=0 (a,I,s)) . k) by A3, AMI_1:123;
IC (((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))) = IC ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) by A2, FUNCT_4:14
.= IC ((StepWhile=0 (a,I,s)) . k) by A1, COMPOS_1:142 ;
hence ((StepWhile=0 (a,I,s)) . k) +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))) = (StepWhile=0 (a,I,s)) . k by A4, Th29; :: thesis: (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3)))
hence (StepWhile=0 (a,I,s)) . (k + 1) = Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),((StepWhile=0 (a,I,s)) . k),((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3)) by Def4, T
.= Comput ((ProgramPart (s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA))))),(s +* ((while=0 (a,I)) +* (Start-At (0,SCM+FSA)))),(n + ((LifeSpan ((ProgramPart (((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA))))),(((StepWhile=0 (a,I,s)) . k) +* (I +* (Start-At (0,SCM+FSA)))))) + 3))) by A3, EXTPRO_1:5 ;
:: thesis: verum