let R be non trivial Ring; :: thesis: for s1, s2 being State of (SCM R) st IC s1 = IC s2 & ( for a being Data-Location of R holds s1 . a = s2 . a ) holds
s1 = s2

let s1, s2 be State of (SCM R); :: thesis: ( IC s1 = IC s2 & ( for a being Data-Location of R holds s1 . a = s2 . a ) implies s1 = s2 )
assume A1: IC s1 = IC s2 ; :: thesis: ( ex a being Data-Location of R st not s1 . a = s2 . a 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 R))) & s2 = (DataPart s2) +* (Start-At ((IC s2),(SCM R))) ) by MEMSTR_0:26;
assume A3: for a being Data-Location of R holds s1 . a = s2 . a ; :: thesis: s1 = s2
DataPart s1 = DataPart s2
proof
A4: 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 dom (DataPart s1) or (DataPart s1) . b1 = (DataPart s2) . b1 )

let x be object ; :: thesis: ( not x in dom (DataPart s1) or (DataPart s1) . x = (DataPart s2) . x )
assume A5: x in dom (DataPart s1) ; :: thesis: (DataPart s1) . x = (DataPart s2) . x
then A6: x is Data-Location of R by ;
thus (DataPart s1) . x = s1 . x by
.= s2 . x by A6, A3
.= (DataPart s2) . x by ; :: thesis: verum
end;
hence s1 = s2 by A1, A2; :: thesis: verum