let s1, s2 be State of SCM+FSA; :: thesis: ( IC s1 = IC s2 & ( for a being Int-Location holds s1 . a = s2 . a ) & ( for f being FinSeq-Location holds s1 . f = s2 . f ) implies s1 = s2 )
assume A1: IC s1 = IC s2 ; :: thesis: ( ex a being Int-Location st not s1 . a = s2 . a or ex f being FinSeq-Location st not s1 . f = s2 . f or s1 = s2 )
( IC in dom s1 & IC in dom s2 ) by MEMSTR_0:2;
then A2: ( s1 = (DataPart s1) +* (Start-At ((IC s1),SCM+FSA)) & s2 = (DataPart s2) +* (Start-At ((IC s2),SCM+FSA)) ) by MEMSTR_0:26;
assume that
A3: for a being Int-Location holds s1 . a = s2 . a and
A4: for f being FinSeq-Location holds s1 . f = s2 . f ; :: thesis: s1 = s2
DataPart s1 = DataPart s2

proof

A5: dom (DataPart s1) = Data-Locations by MEMSTR_0:9;
hence dom (DataPart s1) = dom (DataPart s2) by MEMSTR_0:9; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in proj1 (DataPart s1) or (DataPart s1) . b₁ = (DataPart s2) . b₁ )
let x be object ; :: thesis: ( not x in proj1 (DataPart s1) or (DataPart s1) . x = (DataPart s2) . x )
assume A6: x in dom (DataPart s1) ; :: thesis: (DataPart s1) . x = (DataPart s2) . x
then A7: x in Int-Locations \/ FinSeq-Locations by A5, Th92;

per cases ( x in Int-Locations or x in FinSeq-Locations ) by A7, XBOOLE_0:def 3;

suppose x in Int-Locations ; :: thesis: (DataPart s1) . x = (DataPart s2) . x

then A8: x is Int-Location by AMI_2:def 16;
thus (DataPart s1) . x = s1 . x by A6, A5, FUNCT_1:49
.= s2 . x by A8, A3
.= (DataPart s2) . x by A6, A5, FUNCT_1:49 ; :: thesis: verum

end;

suppose x in FinSeq-Locations ; :: thesis: (DataPart s1) . x = (DataPart s2) . x

then A9: x is FinSeq-Location by Def3;
thus (DataPart s1) . x = s1 . x by A6, A5, FUNCT_1:49
.= s2 . x by A9, A4
.= (DataPart s2) . x by A6, A5, FUNCT_1:49 ; :: thesis: verum

end;

end;

end;

hence s1 = s2 by A1, A2; :: thesis: verum