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),s = (IExec I,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,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),s = (IExec I,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,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),s = (IExec I,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) ) )
if>0 a,J,I is parahalting by Th26;
hence if<0 a,I,J is parahalting by Th20; :: thesis: ( ( s . a < 0 implies IExec (if<0 a,I,J),s = (IExec I,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) ) )

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

A1: I is_halting_on Initialized s by SCMFSA7B:25;
A2: I is_closed_on Initialized s by SCMFSA7B:24;
assume s . a < 0 ; :: thesis: IExec (if<0 a,I,J),s = (IExec I,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA )
hence IExec (if<0 a,I,J),s = (IExec I,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) by A2, A1, Th29; :: thesis: verum

end;

hereby :: thesis: verum

assume A3: s . a >= 0 ; :: thesis: IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA )
A4: J is_closed_on Initialized s by SCMFSA7B:24;
A5: J is_halting_on Initialized s by SCMFSA7B:25;

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

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

hence IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) by A4, A5, Th31; :: thesis: verum

end;

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

hence IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At ((((card I) + (card J)) + (card J)) + 7),SCM+FSA ) by A3, A4, A5, Th33; :: thesis: verum

end;

end;

end;