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
  min(f,g ++ h) c= min(f,g) ++ min(f,h)
proof
  let c;
A1: (min(f,g) ++ min(f,h)).c = min(f,g).c + min(f,h).c - (min(f,g).c)*(min(f
  ,h).c) by Def3
    .= min(f.c,g.c) + min(f,h).c - (min(f,g).c)*(min(f,h).c) by FUZZY_1:5
    .= min(f.c,g.c) + min(f.c,h.c) - (min(f,g).c)*(min(f,h).c) by FUZZY_1:5
    .= min(f.c,g.c) + min(f.c,h.c) - (min(f.c,g.c))*(min(f,h).c) by FUZZY_1:5
    .= min(f.c,g.c) + min(f.c,h.c) - (min(f.c,g.c))*(min(f.c,h.c)) by FUZZY_1:5
;
A2: min(f.c,1 - ((1 - g.c)*(1 - h.c))) <= min(f.c,g.c) + min(f.c,h.c) - (min
  (f.c,g.c))*(min(f.c,h.c))
  proof
    now
      per cases by XXREAL_0:15;
      suppose
A3:     min(f.c,g.c) = f.c & min(f.c,h.c) = f.c;
        f c= f ++ f by Th28;
        then
A4:     (f ++ f).c >= f.c;
        min(f.c,1 - ((1 - g.c)*(1 - h.c))) <= f.c by XXREAL_0:17;
        then min(f.c,1 - ((1 - g.c)*(1 - h.c))) <= (f ++ f).c by A4,XXREAL_0:2;
        hence thesis by A3,Def3;
      end;
      suppose
A5:     min(f.c,g.c) = f.c & min(f.c,h.c) = h.c;
        (1_minus f).c >= 0 by Th1;
        then
A6:     1 - f.c >= 0 by FUZZY_1:def 5;
        h.c >= 0 by Th1;
        then 0*(h.c) <= (h.c)*(1-f.c) by A6,XREAL_1:64;
        then
A7:     0 + f.c <= (h.c)*(1-f.c) + f.c by XREAL_1:6;
        min(f.c,1 - ((1 - g.c)*(1 - h.c))) <= f.c by XXREAL_0:17;
        hence thesis by A5,A7,XXREAL_0:2;
      end;
      suppose
A8:     min(f.c,g.c) = g.c & min(f.c,h.c) = f.c;
        (1_minus f).c >= 0 by Th1;
        then
A9:     1 - f.c >= 0 by FUZZY_1:def 5;
        g.c >= 0 by Th1;
        then 0*(g.c) <= (g.c)*(1-f.c) by A9,XREAL_1:64;
        then
A10:    0 + f.c <= (g.c)*(1-f.c) + f.c by XREAL_1:6;
        min(f.c,1 - ((1 - g.c)*(1 - h.c))) <= f.c by XXREAL_0:17;
        hence thesis by A8,A10,XXREAL_0:2;
      end;
      suppose
        min(f.c,g.c) = g.c & min(f.c,h.c) = h.c;
        hence thesis by XXREAL_0:17;
      end;
    end;
    hence thesis;
  end;
  min(f,g ++ h).c = min(f.c,(g ++ h).c) by FUZZY_1:5
    .= min(f.c,1 - ((1 - g.c)*(1 - h.c))) by Th49;
  hence thesis by A1,A2;
end;
