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 ++ min(g,h) = min((f ++ g),(f ++ h))
proof
A1: C = dom min((f ++ g),(f ++ h)) by FUNCT_2:def 1;
A2: for c being Element of C st c in C holds (f ++ min(g,h)).c = (min((f ++
  g),(f ++ h))).c
  proof
    let c;
A3: (f ++ min(g,h)).c = f.c + min(g,h).c - (f.c)*(min(g,h).c) by Def3
      .= f.c + min(g.c,h.c) - (f.c)*(min(g,h).c) by FUZZY_1:5;
    now
      per cases by XXREAL_0:15;
      suppose
A4:     min(g.c,h.c) = g.c;
A5:     (1_minus f).c >= 0 by Th1;
        g.c <= h.c by A4,XXREAL_0:def 9;
        then (g.c)*((1_minus f).c) <= (h.c)*((1_minus f).c) by A5,XREAL_1:64;
        then (g.c)*(1- f.c) <= (h.c)*((1_minus f).c) by FUZZY_1:def 5;
        then (g.c)*(1- f.c) <= (h.c)*(1- f.c) by FUZZY_1:def 5;
        then (g.c)*1- (g.c)*f.c + f.c <= (h.c)*(1- f.c) + f.c by XREAL_1:6;
        then
A6:     f.c + g.c - (f.c)*g.c = min((f.c + g.c - (f.c)*g.c),(f.c + h.c -
        (f.c)*(h.c))) by XXREAL_0:def 9
          .= min((f ++ g).c,(f.c + h.c - (f.c)*(h.c))) by Def3
          .= min((f ++ g).c,(f ++ h).c) by Def3;
        (f ++ min(g,h)).c = f.c + g.c - (f.c)*g.c by A3,A4,FUZZY_1:5;
        hence thesis by A6,FUZZY_1:5;
      end;
      suppose
A7:     min(g.c,h.c) = h.c;
A8:     (1_minus f).c >= 0 by Th1;
        h.c <= g.c by A7,XXREAL_0:def 9;
        then (h.c)*((1_minus f).c) <= (g.c)*((1_minus f).c) by A8,XREAL_1:64;
        then (h.c)*(1- f.c) <= (g.c)*((1_minus f).c) by FUZZY_1:def 5;
        then (h.c)*(1- f.c) <= (g.c)*(1- f.c) by FUZZY_1:def 5;
        then (h.c)*1- (h.c)*f.c + f.c <= (g.c)*(1- f.c) + f.c by XREAL_1:6;
        then
A9:     f.c + h.c - (f.c)*h.c = min((f.c + g.c - (f.c)*g.c),(f.c + h.c -
        (f.c)*(h.c))) by XXREAL_0:def 9
          .= min((f ++ g).c,(f.c + h.c - (f.c)*(h.c))) by Def3
          .= min((f ++ g).c,(f ++ h).c) by Def3;
        (f ++ min(g,h)).c = f.c + h.c - (f.c)*h.c by A3,A7,FUZZY_1:5;
        hence thesis by A9,FUZZY_1:5;
      end;
    end;
    hence thesis;
  end;
  C = dom (f ++ min(g,h)) by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
