theorem Th6:
  x in dom F & v = F.x implies (f (#) F).x = f.v * v
proof
  assume that
A1: x in dom F and
A2: v = F.x;
A3: F/.x = F.x by A1,PARTFUN1:def 6;
  len(f (#) F) = len F by Def5;
  then x in dom(f (#) F) by A1,FINSEQ_3:29;
  hence thesis by A2,A3,Def5;
end;
