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 ++ h) c= max(f,g) ++ max(f,h)
proof
  let c;
A1: (max(f,g) ++ max(f,h)).c = max(f,g).c + max(f,h).c - (max(f,g).c)*(max(f
  ,h).c) by Def3
    .= max(f.c,g.c) + max(f,h).c - (max(f,g).c)*(max(f,h).c) by FUZZY_1:5
    .= max(f.c,g.c) + max(f.c,h.c) - (max(f,g).c)*(max(f,h).c) by FUZZY_1:5
    .= max(f.c,g.c) + max(f.c,h.c) - (max(f.c,g.c))*(max(f,h).c) by FUZZY_1:5
    .= max(f.c,g.c) + max(f.c,h.c) - (max(f.c,g.c))*(max(f.c,h.c)) by FUZZY_1:5
;
A2: max(f.c,1 - ((1 - g.c)*(1 - h.c))) <= max(f.c,g.c) + max(f.c,h.c) - (max
  (f.c,g.c))*(max(f.c,h.c))
  proof
    per cases by XXREAL_0:16;
    suppose
A3:   max(f.c,g.c) = f.c & max(f.c,h.c) = f.c;
      (1_minus g).c >= 0 by Th1;
      then
A4:   1-g.c >= 0 by FUZZY_1:def 5;
      h.c <= f.c by A3,XXREAL_0:def 10;
      then 1 - h.c >= 1 - f.c by XREAL_1:10;
      then
A5:   (1 - g.c)*(1 - f.c) <= (1 - g.c)*(1 - h.c) by A4,XREAL_1:64;
      (1_minus f).c >= 0 by Th1;
      then
A6:   1-f.c >= 0 by FUZZY_1:def 5;
      f c= f++f by Th28;
      then (f ++ f).c >= f.c;
      then
A7:   f.c + f.c - (f.c)*(f.c) >= f.c by Def3;
      g.c <= f.c by A3,XXREAL_0:def 10;
      then 1 - g.c >= 1 - f.c by XREAL_1:10;
      then (1 - f.c)*(1 - f.c) <= (1 - g.c)*(1 - f.c) by A6,XREAL_1:64;
      then (1 - f.c)*(1 - f.c) <= (1 - g.c)*(1 - h.c) by A5,XXREAL_0:2;
      then 1 - (1 - f.c)*(1 - f.c) >= 1 - (1 - g.c)*(1 - h.c) by XREAL_1:10;
      hence thesis by A3,A7,XXREAL_0:28;
    end;
    suppose
A8:   max(f.c,g.c) = f.c & max(f.c,h.c) = h.c;
      (1_minus f).c >= 0 by Th1;
      then
A9:   1 - f.c >= 0 by FUZZY_1:def 5;
      h.c >= 0 by Th1;
      then 0*(h.c) <= (h.c)*(1-f.c) by A9,XREAL_1:64;
      then
A10:  0 + f.c <= (h.c)*(1-f.c) + f.c by XREAL_1:6;
      (1_minus h).c >= 0 by Th1;
      then
A11:  1 - h.c >= 0 by FUZZY_1:def 5;
      g.c <= f.c by A8,XXREAL_0:def 10;
      then 1 - f.c <= 1 - g.c by XREAL_1:10;
      then (1 - f.c)*(1 -h.c) <= (1 - g.c)*(1 -h.c) by A11,XREAL_1:64;
      then 1-(1 - f.c)*(1 -h.c) >= 1-(1 - g.c)*(1 -h.c) by XREAL_1:10;
      hence thesis by A8,A10,XXREAL_0:28;
    end;
    suppose
A12:  max(f.c,g.c) = g.c & max(f.c,h.c) = f.c;
      (1_minus f).c >= 0 by Th1;
      then
A13:  1 - f.c >= 0 by FUZZY_1:def 5;
      g.c >= 0 by Th1;
      then 0*(g.c) <= (g.c)*(1-f.c) by A13,XREAL_1:64;
      then
A14:  0 + f.c <= (g.c)*(1-f.c) + f.c by XREAL_1:6;
      (1_minus g).c >= 0 by Th1;
      then
A15:  1 - g.c >= 0 by FUZZY_1:def 5;
      h.c <= f.c by A12,XXREAL_0:def 10;
      then 1 - f.c <= 1 - h.c by XREAL_1:10;
      then (1 - f.c)*(1 -g.c) <= (1 - h.c)*(1 -g.c) by A15,XREAL_1:64;
      then 1-(1 - f.c)*(1 -g.c) >= 1-(1 - h.c)*(1 - g.c) by XREAL_1:10;
      hence thesis by A12,A14,XXREAL_0:28;
    end;
    suppose
A16:  max(f.c,g.c) = g.c & max(f.c,h.c) = h.c;
      (1_minus g).c >= 0 by Th1;
      then
A17:  1-g.c >= 0 by FUZZY_1:def 5;
      h.c >= f.c by A16,XXREAL_0:def 10;
      then 1 - h.c <= 1 - f.c by XREAL_1:10;
      then
A18:  (1 - g.c)*(1 - f.c) >= (1 - g.c)*(1 - h.c) by A17,XREAL_1:64;
      (1_minus f).c >= 0 by Th1;
      then
A19:  1-f.c >= 0 by FUZZY_1:def 5;
      g.c >= f.c by A16,XXREAL_0:def 10;
      then 1 - g.c <= 1 - f.c by XREAL_1:10;
      then (1 - f.c)*(1 - f.c) >= (1 - g.c)*(1 - f.c) by A19,XREAL_1:64;
      then (1 - f.c)*(1 - f.c) >= (1 - g.c)*(1 - h.c) by A18,XXREAL_0:2;
      then 1 - ((1 - f.c)*(1 - f.c)) <= 1 - (1 - g.c)*(1 - h.c) by XREAL_1:10;
      then
A20:  (f ++ f).c <= 1 - (1 - g.c)*(1 - h.c) by Th49;
      f c= f++f by Th28;
      then (f ++ f).c >= f.c;
      then f.c <= 1 - (1 - g.c)*(1 - h.c) by A20,XXREAL_0:2;
      hence thesis by A16,XXREAL_0:28;
    end;
  end;
  max(f,g ++ h).c = max(f.c,(g ++ h).c) by FUZZY_1:5
    .= max(f.c,1 - ((1 - g.c)*(1 - h.c))) by Th49;
  hence thesis by A1,A2;
end;
