reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem Th24:
  F is associative implies F[;](d,id D)*(F.:(f,f9)) = F.:(F[;](d, id D)*f,f9)
proof
  assume
A1: F is associative;
  now
    let c;
    thus (F[;](d,id D)*(F.:(f,f9))).c = (F[;](d,id D)).((F.:(f,f9)).c) by
FUNCT_2:15
      .= (F[;](d,id D)).(F.(f.c,f9.c)) by FUNCOP_1:37
      .= F.(d,(id D).(F.(f.c,f9.c))) by FUNCOP_1:53
      .= F.(d,F.(f.c,f9.c))
      .= F.(F.(d,f.c),f9.c) by A1
      .= F.((F[;](d,f)).c,f9.c) by FUNCOP_1:53
      .= F.(((F[;](d,id D))*f).c,f9.c) by FUNCOP_1:55
      .= (F.:(F[;](d,id D)*f,f9)).c by FUNCOP_1:37;
  end;
  hence thesis by FUNCT_2:63;
end;
