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(1_minus f, 1_minus g)) c= 1_minus (f \+\ g)
proof
  set f1 = 1_minus f, g1 = 1_minus g;
  let x be Element of C;
  1_minus (f \+\ g) = min( 1_minus min(f,g1) , 1_minus min(f1,g)) by Th10
    .= min( max(f1,1_minus g1),1_minus min(f1,g)) by Th10
    .= min( max(f1,g),max(1_minus f1,g1)) by Th10
    .= max(min(max(f1,g),f),min(max(f1,g),g1)) by Th9
    .= max( max(min(f,f1),min(f,g)) ,min(max(f1,g),g1)) by Th9
    .= max( max(min(f,f1),min(f,g)) , max(min(g1,f1),min(g1,g))) by Th9
    .= max( max(max(min(f,g),min(f,f1)),min(g1,f1)) , min(g1,g)) by Th7
    .= max( max(max(min(g1,f1),min(f,g)),min(f,f1)) , min(g1,g)) by Th7
    .= max( max(min(g1,f1),min(f,g)), max(min(f,f1) , min(g1,g))) by Th7;
  then
  (1_minus (f \+\ g)).x = max(max(min(f,g),min(f1,g1)).x,max(min(f,f1),min
  (g1,g)).x) by Def4;
  hence thesis by XXREAL_0:25;
end;
