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, min(f,g)) c= max(f,g)
proof
  set f1 = 1_minus f, g1 = 1_minus g;
  let x be Element of C;
  max(f \+\ g, min(f,g)) = max(min(f,g1),max(min(f1,g),min(f,g))) by Th7
    .= max(min(f,g1),min(max(min(f1,g),f),max(g,min(f1,g)))) by Th9
    .= max(min(f,g1),min(max(min(f1,g),f),g)) by Th8
    .= min(max(min(f,g1),max(f,min(f1,g))),max(min(f,g1),g)) by Th9
    .= min(max(max(f,min(f,g1)),min(f1,g)) ,max(min(f,g1),g)) by Th7
    .= min(max(f,min(f1,g)) ,max(min(f,g1),g)) by Th8
    .= min( min(max(f,f1),max(f,g)) ,max(g,min(f,g1))) by Th9
    .= min( min(max(f,f1),max(f,g)) , min(max(g,f),max(g,g1)) ) by Th9
    .= min(min( min(max(f,f1),max(f,g)),max(g,f) ),max(g,g1) ) by Th7
    .= min(min( max(f,f1),min(max(f,g),max(f,g)) ),max(g,g1) ) by Th7
    .= min(max(f,g),min(max(f,f1),max(g,g1))) by Th7;
  then
  max(f \+\ g, min(f,g)).x = min(max(f,g).x,min(max(f,f1),max(g,g1)).x) by Def3
;
  hence thesis by XXREAL_0:17;
end;
