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)*min(f,h) c= min(f,(g*h))
proof
  let c;
A1: min(f.c,g.c)*min(f.c,h.c) <= min(f,(g*h)).c
  proof
    now
      per cases by XXREAL_0:15;
      suppose
A2:     min(f.c,g.c) = f.c;
        min(f.c,g.c)*min(f.c,h.c) <= min(f,(g*h)).c
        proof
          per cases by XXREAL_0:15;
          suppose
A3:         min(f.c,h.c) = f.c;
A4:         0 <= g.c by Th1;
            f.c <= h.c by A3,XXREAL_0:def 9;
            then
A5:         (g.c)*(f.c) <= (h.c)*(g.c) by A4,XREAL_1:64;
A6:         0 <= f.c by Th1;
            f*f c= f by Th28;
            then
A7:         (f*f).c <= f.c;
            f.c <= g.c by A2,XXREAL_0:def 9;
            then (f.c)*(f.c) <= (g.c)*(f.c) by A6,XREAL_1:64;
            then (f.c)*(f.c) <= (g.c)*(h.c) by A5,XXREAL_0:2;
            then min((f*f).c,(f.c)*(f.c)) <= min(f.c,(g.c)*(h.c)) by A7,
XXREAL_0:18;
            then min((f.c)*(f.c),(f.c)*(f.c)) <= min(f.c,(g.c)*(h.c)) by Def2;
            then (f.c)*(f.c) <= min(f.c,(g*h).c) by Def2;
            hence thesis by A2,A3,FUZZY_1:5;
          end;
          suppose
A8:         min(f.c,h.c) = h.c;
A9:         1 >= h.c by Th1;
A10:        h.c >= 0 by Th1;
            f.c <= g.c by A2,XXREAL_0:def 9;
            then
A11:        (f.c)*(h.c) <= (g.c)*(h.c) by A10,XREAL_1:64;
            f.c >= 0 by Th1;
            then (f.c)*(h.c) <= (f.c)*1 by A9,XREAL_1:64;
            then (f.c)*(h.c) <= min((f.c),(g.c)*(h.c)) by A11,XXREAL_0:20;
            then (f.c)*(h.c) <= min(f.c,(g*h).c) by Def2;
            hence thesis by A2,A8,FUZZY_1:5;
          end;
        end;
        hence thesis;
      end;
      suppose
A12:    min(f.c,g.c) = g.c;
        min(f.c,g.c)*min(f.c,h.c) <= min(f,(g*h)).c
        proof
          per cases by XXREAL_0:15;
          suppose
A13:        min(f.c,h.c) = f.c;
A14:        g.c >= 0 by Th1;
            f.c <= h.c by A13,XXREAL_0:def 9;
            then
A15:        (f.c)*(g.c) <= (h.c)*(g.c) by A14,XREAL_1:64;
A16:        1 >= g.c by Th1;
            f.c >= 0 by Th1;
            then (g.c)*(f.c) <= (f.c)*1 by A16,XREAL_1:64;
            then (f.c)*(g.c) <= min((f.c),(h.c)*(g.c)) by A15,XXREAL_0:20;
            then (f.c)*(g.c) <= min(f.c,(g*h).c) by Def2;
            hence thesis by A12,A13,FUZZY_1:5;
          end;
          suppose
A17:        min(f.c,h.c) = h.c;
A18:        g.c <= f.c by A12,XXREAL_0:def 9;
            f.c >= 0 by Th1;
            then
A19:        (g.c)*(f.c) <= (f.c)*(f.c) by A18,XREAL_1:64;
            f*f c= f by Th28;
            then (f*f).c <= f.c;
            then
A20:        (f.c)*(f.c) <= f.c by Def2;
A21:        h.c <= f.c by A17,XXREAL_0:def 9;
            g.c >= 0 by Th1;
            then (h.c)*(g.c) <= (f.c)*(g.c) by A21,XREAL_1:64;
            then (h.c)*(g.c) <= (f.c)*(f.c) by A19,XXREAL_0:2;
            then (h.c)*(g.c) <= f.c by A20,XXREAL_0:2;
            then (h.c)*(g.c) <= min(f.c,(h.c)*(g.c)) by XXREAL_0:20;
            then (g.c)*(h.c) <= min(f.c,(g*h).c) by Def2;
            hence thesis by A12,A17,FUZZY_1:5;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  (min(f,g)*min(f,h)).c = (min(f,g).c)*(min(f,h).c) by Def2
    .= min(f.c,g.c)*(min(f,h).c) by FUZZY_1:5
    .= min(f.c,g.c)*min(f.c,h.c) by FUZZY_1:5;
  hence thesis by A1;
end;
