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)) = EMF(C) & f*(UMF(C)) = f
proof
A1: C = dom (f*EMF(C)) by FUNCT_2:def 1;
A2: C = dom f by FUNCT_2:def 1;
A3: for c being Element of C st c in C holds (f*(UMF(C))).c = f.c
  proof
    let c;
    (f*(UMF(C))).c = (f.c)*((UMF(C)).c) by Def2
      .= (f.c)*1 by FUNCT_3:def 3;
    hence thesis;
  end;
A4: for c being Element of C st c in C holds (f*(EMF C)).c = (EMF C).c
  proof
    let c;
    (f*EMF(C)).c = (f.c)*((EMF C).c) by Def2
      .= (f.c)*0 by FUNCT_3:def 3;
    hence thesis by FUNCT_3:def 3;
  end;
A5: C = dom (f*UMF C) by FUNCT_2:def 1;
  C = dom EMF(C) by FUNCT_2:def 1;
  hence thesis by A1,A4,A2,A5,A3,PARTFUN1:5;
end;
