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= max(g,h) & min(f,h) = EMF(C) implies f c= g
proof
  assume that
A1: f c= max(g,h) and
A2: min(f,h) = EMF(C);
  let x be Element of C;
  min(f,max(g,h)) = f by A1,Th26;
  then f = max(min(f,g),min(f,h)) by Th9
    .= min(f,g) by A2,Th17;
  then f.x = min(f.x,g.x) by Def3;
  hence thesis by XXREAL_0:def 9;
end;
