let p be Instruction-Sequence of SCM+FSA; :: thesis: for s being State of SCM+FSA
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)),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 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)),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 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)),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 InitHalting & ( 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,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)),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)) ) )
( I is_closed_onInit s,p & I is_halting_onInit s,p ) by Th35, Th36;
hence ( 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)) ) by Th55; :: 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)) )
A1: ( J is_closed_onInit s,p & J is_halting_onInit s,p ) by Th35, Th36;

hereby :: thesis: verum

assume A2: 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))

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

end;

end;

end;