let N be with_zero set ; :: thesis: for S being 1 -element with_non-empty_values IC-Ins-separated Mem-Struct over N
for s1, s2 being State of S st IC s1 = IC s2 holds
s1 = s2
let T be 1 -element with_non-empty_values IC-Ins-separated Mem-Struct over N; :: thesis: for s1, s2 being State of T st IC s1 = IC s2 holds
s1 = s2
let s1, s2 be State of T; :: thesis: ( IC s1 = IC s2 implies s1 = s2 )
assume A1: IC s1 = IC s2 ; :: thesis: s1 = s2
A2: dom s1 = the carrier of T by PARTFUN1:def 2;
then A3: dom s1 = dom s2 by PARTFUN1:def 2;

now :: thesis: for x being object st x in dom s1 holds
s1 . x = s2 . x

let x be object ; :: thesis: ( x in dom s1 implies s1 . x = s2 . x )
assume A4: x in dom s1 ; :: thesis: s1 . x = s2 . x
A5: x = IC by A4, A2, STRUCT_0:def 10;
hence s1 . x = IC s1
.= s2 . x by A1, A5 ;
:: thesis: verum

end;

hence s1 = s2 by A3; :: thesis: verum