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 Th29:
  (f*g)*h = f*(g*h)
proof
A1: C = dom (f*(g*h)) by FUNCT_2:def 1;
A2: for c being Element of C st c in C holds ((f*g)*h).c = (f*(g*h)).c
  proof
    let c;
    ((f*g)*h).c = ((f*g).c)*(h.c) by Def2
      .= ((f.c)*(g.c))*(h.c) by Def2
      .= (f.c)*((g.c)*(h.c))
      .= (f.c)*((g*h).c) by Def2;
    hence thesis by Def2;
  end;
  C = dom ((f*g)*h) by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
