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 (insloc ((((card I) + (card J)) + (card J)) + 7))) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) ) )
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 (insloc ((((card I) + (card J)) + (card J)) + 7))) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) ) )
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 (insloc ((((card I) + (card J)) + (card J)) + 7))) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) ) )
A1: if<0 a,I,J = if=0 a,J,(if>0 a,J,I) by SCMFSA8B:def 3;
if>0 a,J,I is InitHalting by Th53;
hence if<0 a,I,J is InitHalting by A1, Th47; :: thesis: ( ( s . a < 0 implies IExec (if<0 a,I,J),s = (IExec I,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) ) & ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) ) )
A2: ( I is_closed_onInit s & I is_halting_onInit s ) by Th35, Th36;
A3: ( J is_closed_onInit s & J is_halting_onInit s ) by Th35, Th36;
thus ( s . a < 0 implies IExec (if<0 a,I,J),s = (IExec I,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) ) by A2, Th55; :: thesis: ( s . a >= 0 implies IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) )

hereby :: thesis: verum

assume A4: s . a >= 0 ; :: thesis: IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7)))

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 (insloc ((((card I) + (card J)) + (card J)) + 7)))

hence IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) by A3, Th56; :: thesis: verum

end;

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

hence IExec (if<0 a,I,J),s = (IExec J,s) +* (Start-At (insloc ((((card I) + (card J)) + (card J)) + 7))) by A3, A4, Th57; :: thesis: verum

end;

end;

end;