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 c= g & f c= h & min(g,h) = EMF(C) implies f = EMF(C)
proof
  assume that
A1: f c= g & f c= h and
A2: min(g,h)= EMF(C);
A3: for x being Element of C st x in C holds f.x = (EMF(C)).x
  proof
    let x be Element of C;
    f.x <= g.x & f.x <= h.x by A1;
    then f.x <= min(g.x,h.x) by XXREAL_0:20;
    then
A4: f.x <= min(g,h).x by Def3;
    (EMF(C)).x <= f.x by Th15;
    hence thesis by A2,A4,XXREAL_0:1;
  end;
  C = dom f & C = dom EMF(C) by FUNCT_2:def 1;
  hence thesis by A3,PARTFUN1:5;
end;
