theorem Th50:
  i in dom (a*p) & a9 = p.i implies (a*p).i = a*a9
proof
  assume
A1: i in dom (a*p) & a9 = p.i; then
A2: a9 in dom((the multF of K)[;](a,id the carrier of K)) by FUNCT_1:11;
  thus (a*p).i = ((the multF of K)[;](a,id the carrier of K)).a9 by A1,
FUNCT_1:12
    .=(the multF of K).(a,(id the carrier of K).a9) by A2,FUNCOP_1:32
    .= a*a9;
end;
