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*max(g,h) = max(f*g,f*h)
proof
A1: C = dom max(f*g,f*h) by FUNCT_2:def 1;
A2: for c being Element of C st c in C holds (f*max(g,h)).c = max(f*g,f*h).c
  proof
    let c;
    (f*max(g,h)).c = (f.c)*(max(g,h).c) by Def2
      .= (f.c)*max(g.c,h.c) by FUZZY_1:5
      .= max((f.c)*g.c,(f.c)*(h.c)) by Lm2,Th1
      .= max((f*g).c,(f.c)*(h.c)) by Def2
      .= max((f*g).c,(f*h).c) by Def2;
    hence thesis by FUZZY_1:5;
  end;
  C = dom (f*max(g,h)) by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
