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) c= f ++ g
proof
  let c;
A1: max(f,g).c = max(f.c,g.c) by FUZZY_1:5;
  per cases by A1,XXREAL_0:16;
  suppose
A2: max(f,g).c = f.c;
A3: (1_minus f).c >= 0 by Th1;
    g.c >= 0 by Th1;
    then 0*(g.c) <= (g.c)*((1_minus f).c) by A3,XREAL_1:64;
    then 0 <= (g.c)*(1 - f.c) by FUZZY_1:def 5;
    then 0 + f.c <= g.c - (f.c)*(g.c) + f.c by XREAL_1:6;
    then f.c <= f.c + g.c - (f.c)*(g.c);
    hence thesis by A2,Def3;
  end;
  suppose
A4: max(f,g).c = g.c;
A5: (1_minus g).c >= 0 by Th1;
    f.c >= 0 by Th1;
    then 0*(f.c) <= (f.c)*((1_minus g).c) by A5,XREAL_1:64;
    then 0 <= (f.c)*(1 - g.c) by FUZZY_1:def 5;
    then 0 + g.c <= f.c - (f.c)*(g.c) + g.c by XREAL_1:6;
    then g.c <= f.c + g.c - (f.c)*(g.c);
    hence thesis by A4,Def3;
  end;
end;
