let s be State of SCM+FSA ; :: thesis: for I, J being InitHalting Program of SCM+FSA
for a being read-write Int-Location holds
( if<0 a,I,J is InitHalting & ( 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 InitHalting Program of SCM+FSA ; :: thesis: for a being read-write Int-Location holds
( if<0 a,I,J is InitHalting & ( 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 InitHalting & ( 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,I,J = if=0 a,J,(if>0 a,J,I) & if>0 a,J,I is InitHalting ) by Th53, SCMFSA8B:def 3;
hence if<0 a,I,J is InitHalting by Th47; :: 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 ) ) )
( I is_closed_onInit s & I is_halting_onInit s ) by Th35, Th36;
hence ( 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 ) ) by Th55; :: 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: ( J is_closed_onInit s & J is_halting_onInit s ) by Th35, Th36;

hereby :: thesis: verum

assume A2: 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 )

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 A1, Th56; :: 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 A1, A2, Th57; :: thesis: verum

end;

end;

end;