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 \+\ g)\h = max(f\max(g,h),g\max(f,h))
proof
  set f1 = 1_minus f, g1 = 1_minus g, h1 = 1_minus h;
  max(min(f,g1),min(f1,g))\h = max(min(h1,min(f,g1)),min(h1,min(f1,g))) by
FUZZY_1:9
    .= max(min(min(h1,g1),f),min(h1,min(f1,g))) by FUZZY_1:7
    .= max(min(min(h1,g1),f),min(min(h1,f1),g)) by FUZZY_1:7
    .= max(min(f,1_minus max(h,g)),min(g,min(h1,f1))) by FUZZY_1:11
    .= max(f\max(h,g),g\max(f,h)) by FUZZY_1:11;
  hence thesis;
end;
