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 c;
  max(f \+\ g,min(f,g)) = max(min(f,g1),max(min(f1,g),min(f,g))) by FUZZY_1:7
    .= max(min(f,g1),min(max(min(f1,g),f),max(min(f1,g),g))) by FUZZY_1:9
    .= max(min(f,g1),min(max(min(f1,g),f),g)) by FUZZY_1:8
    .= min(max(min(f,g1),max(min(f1,g),f)),max(min(f,g1),g)) by FUZZY_1:9;
  then
  max(f \+\ g,min(f,g)).c = min(max(min(f,g1),max(min(f1,g),f)).c,max(min(
  f,g1),g).c) by FUZZY_1:5;
  then
A1: max(f \+\ g, min(f,g)).c <= max(min(f,g1),g).c by XXREAL_0:17;
  max(min(f,g1),g).c = min(max(g,f),max(g,g1)).c by FUZZY_1:9
    .= min(max(f,g).c,max(g,g1).c) by FUZZY_1:5;
  then max(min(f,g1),g).c <= max(f,g).c by XXREAL_0:17;
  hence thesis by A1,XXREAL_0:2;
end;
