let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: 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 Th26;
hence if<0 (a,I,J) is parahalting by Th20; :: 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)) ) )

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)) )

A1: I is_halting_on Initialized s,P by SCMFSA7B:25;
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 A1, Th29, SCMFSA7B:24; :: thesis: verum

end;

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))
A3: J is_halting_on Initialized s,P by SCMFSA7B:25;

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 A3, Th31, SCMFSA7B:24; :: 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 A2, A3, Th33, SCMFSA7B:24; :: thesis: verum

end;

end;

end;