theorem Th7:
  g is commutative associative & (g is having_a_unity or len F >= 1
  ) & G = F * P implies g "**" F = g "**" G
proof
  assume that
A1: g is commutative associative and
A2: g is having_a_unity or len F >= 1;
  assume
A3: G = F * P;
  now
    per cases;
    suppose
      len F = 0;
      hence thesis by A2,A3,Lm8;
    end;
    suppose
A4:   len F <> 0;
      len F = len G & for i st i in dom G holds G.i = F.(P.i) by A3,FINSEQ_2:44
,FUNCT_1:12;
      hence thesis by A1,A4,Lm7,NAT_1:14;
    end;
  end;
  hence thesis;
end;
