theorem Th31:
  dom T1 = dom T2 & (for p st p in dom T1 holds T1.p = T2.p) implies T1 = T2
proof
  assume that
A1: dom T1 = dom T2 and
A2: for p st p in dom T1 holds T1.p = T2.p;
 now
    let x be object;
    assume x in dom T1;
    then reconsider p = x as Element of dom T1;
 T1.p = T2.p by A2;
    hence T1.x = T2.x;
  end;
  hence thesis by A1;
end;
