theorem Th4:
  F is with_the_same_dom & E c= dom(F.0) & (for m be Nat holds F.m
  is E-measurable) implies (F||E).n is E-measurable
proof
  set G = F||E;
  assume
A1: F is with_the_same_dom & E c= dom(F.0);
  assume
A2: for m be Nat holds F.m is E-measurable;
  for m be Nat holds (R_EAL F).m is E-measurable & (R_EAL G).m = ((
  R_EAL F).m)|E
  proof
    let m be Nat;
    F.m is E-measurable by A2;
    then R_EAL(F.m) is E-measurable;
    hence (R_EAL F).m is E-measurable;
    thus (R_EAL G).m = ((R_EAL F).m)|E by Def1;
  end;
  then R_EAL(G.n) is E-measurable by A1,MESFUNC9:20;
  hence G.n is E-measurable;
end;
