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 = min(g,h) iff f c= g & f c= h & for h1 st h1 c= g & h1 c= h holds h1 c= f
proof
  hereby
    assume
A1: f = min(g,h);
    hence f c= g & f c= h by Th16;
    let h1;
    assume
A2: h1 c= g & h1 c= h;
    thus h1 c= f
    proof
      let x be Element of C;
      h1.x <= g.x & h1.x <= h.x by A2;
      then min(g.x,h.x) >= h1.x by XXREAL_0:20;
      hence thesis by A1,Def3;
    end;
  end;
  assume that
A3: f c= g & f c= h and
A4: for h1 st h1 c= g & h1 c= h holds h1 c= f;
  min(g,h) c= g & min(g,h) c= h by Th16;
  then
A5: min(g,h) c= f by A4;
  f c= min(g,h) by A3,Th23;
  hence thesis by A5,Lm1;
end;
