theorem Th25:
  Re F is with_the_same_dom iff Im F is with_the_same_dom
proof
  hereby
    assume Re F is with_the_same_dom;
    then
 F is with_the_same_dom by Th24;
    then for n,m being Nat holds dom((Im F).n) = dom((Im F).m)
       by MESFUN7C:def 12,MESFUNC8:def 2;
    hence Im F is with_the_same_dom;
  end;
  assume
A1: Im F is with_the_same_dom;
  now
    let n,m be Nat;
    dom((Im F).n) = dom(F.n) & dom((Im F).m) = dom(F.m) by MESFUN7C:def 12;
    hence dom(F.n) = dom(F.m) by A1;
  end;
  then F is with_the_same_dom;
  hence Re F is with_the_same_dom;
end;
