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(min(f,g),min(f,h)) = f implies f c= max(g,h)
proof
  assume
A1: max(min(f,g),min(f,h)) = f;
  let x be Element of C;
  max(min(f,g),min(f,h)).x = max(min(f,g).x,min(f,h).x) by Def4
    .=max(min(f,g).x,min(f.x,h.x)) by Def3
    .=max(min(f.x,g.x),min(f.x,h.x)) by Def3
    .=min(f.x,max(g.x,h.x)) by XXREAL_0:38;
  then f.x <= max(g.x,h.x) by A1,XXREAL_0:def 9;
  hence thesis by Def4;
end;
