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
  max(f,g) = EMF(C) iff f = EMF(C) & g = EMF(C)
proof
  thus max(f,g) = EMF(C) implies f = EMF(C) & g = EMF(C)
  proof
    assume
A1: max(f,g) = EMF(C);
A2: for x being Element of C st x in C holds f.x = (EMF(C)).x
    proof
      let x be Element of C;
      max(f.x,g.x) =(EMF(C)).x by A1,Def4;
      then
A3:   f.x <= (EMF(C)).x by XXREAL_0:25;
      (EMF(C)).x <= f.x by Th15;
      hence thesis by A3,XXREAL_0:1;
    end;
A4: for x being Element of C st x in C holds g.x = (EMF(C)).x
    proof
      let x be Element of C;
      max(f.x,g.x) =(EMF(C)).x by A1,Def4;
      then
A5:   g.x <= (EMF(C)).x by XXREAL_0:25;
      (EMF(C)).x <= g.x by Th15;
      hence thesis by A5,XXREAL_0:1;
    end;
    C = dom f & C = dom EMF(C) by FUNCT_2:def 1;
    hence f = EMF(C) by A2,PARTFUN1:5;
    C = dom g & C = dom EMF(C) by FUNCT_2:def 1;
    hence g = EMF(C) by A4,PARTFUN1:5;
  end;
  assume f = EMF(C) & g = EMF(C);
  hence thesis;
end;
