let P be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
let s be State of SCM+FSA; :: thesis: for I, J being parahalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
let I, J be parahalting Program of SCM+FSA; :: thesis: for a being read-write Int-Location holds
( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
let a be read-write Int-Location; :: thesis: ( if<0 (a,I,J) is parahalting & ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
if>0 (a,J,I) is parahalting by Th23;
hence if<0 (a,I,J) is parahalting by Th17; :: thesis: ( ( s . a < 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) & ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) ) )
A1: I is_closed_on Initialized s,P by SCMFSA7B:18;

hereby :: thesis: ( s . a >= 0 implies IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) )

A2: I is_halting_on Initialized s,P by SCMFSA7B:19;
assume s . a < 0 ; :: thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
hence IExec ((if<0 (a,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by A2, Th26, A1; :: thesis: verum

end;

hereby :: thesis: verum

assume A3: s . a >= 0 ; :: thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))
A4: J is_halting_on Initialized s,P by SCMFSA7B:19;
A5: J is_closed_on Initialized s,P by SCMFSA7B:18;

per cases ( s . a = 0 or s . a <> 0 ) ;

suppose s . a = 0 ; :: thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))

hence IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by A4, Th28, A5; :: thesis: verum

end;

suppose s . a <> 0 ; :: thesis: IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA))

hence IExec ((if<0 (a,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At (((((card I) + (card J)) + (card J)) + 7),SCM+FSA)) by A3, A4, Th30, A5; :: thesis: verum

end;

end;

end;