reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem
  f ++ EMF(C) = f & f ++ UMF(C) = UMF(C)
proof
A1: C = dom (f ++ EMF(C)) by FUNCT_2:def 1;
A2: C = dom UMF(C) by FUNCT_2:def 1;
A3: for c being Element of C st c in C holds (f ++ UMF(C)).c = (UMF(C)).c
  proof
    let c;
    (f ++ UMF(C)).c = f.c + (UMF(C)).c - (f.c)*((UMF(C)).c) by Def3
      .= (UMF(C)).c+f.c - (f.c)*1 by FUNCT_3:def 3
      .= (UMF(C)).c + (f.c - f.c);
    hence thesis;
  end;
A4: for c being Element of C st c in C holds (f ++ EMF(C)).c = f.c
  proof
    let c;
    (f ++ EMF(C)).c = f.c + (EMF(C)).c - (f.c)*((EMF(C)).c) by Def3
      .= f.c + 0 - (f.c)*((EMF(C)).c) by FUNCT_3:def 3
      .= f.c - (f.c)*0 by FUNCT_3:def 3;
    hence thesis;
  end;
A5: C = dom (f ++ UMF(C)) by FUNCT_2:def 1;
  C = dom f by FUNCT_2:def 1;
  hence thesis by A1,A4,A2,A5,A3,PARTFUN1:5;
end;
