theorem Th25:
  f (#) <* v *> = <* v * f.v *>
proof
A1: 1 in {1} by TARSKI:def 1;
A2: len(f (#) <* v *>) = len<* v *> by Def6
    .= 1 by FINSEQ_1:40;
  then dom(f (#) <* v *>) = {1} by FINSEQ_1:2,def 3;
  then (f (#) <* v *>).1 = (<* v *>/.1) * f.(<* v *>/.1) by A1,Def6
    .= v * f.(<* v *>/.1) by FINSEQ_4:16
    .= v * f.v by FINSEQ_4:16;
  hence thesis by A2,FINSEQ_1:40;
end;
