theorem
  |(f1,(0.REAL n)+*(x,r))| = f1.x * r
  proof
A1: mlt(f1,(0.REAL n)+*(x,r)) = (0.REAL n)+*(x,f1.x*r) by Th15;
A2: dom f1 = Seg n by FINSEQ_1:89;
A3: n in NAT by ORDINAL1:def 12;
    per cases;
    suppose x in dom f1;
      hence thesis by A1,A2,A3,JORDAN2B:10;
    end;
    suppose not x in dom f1;
      then
A4:   f1.x = 0 by FUNCT_1:def 2;
      hence |(f1,(0.REAL n)+*(x,r))| = Sum 0.REAL n by A1,Th14
      .= f1.x * r by A4,A3,JORDAN2B:9;
    end;
  end;
