reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th18:
  mlt(R1,R2).j = R1.j * R2.j
proof
  reconsider i,j1=j as Element of NAT by ORDINAL1:def 12;
  reconsider R1,R2 as Element of i-tuples_on COMPLEX;
  per cases;
  suppose
A1: not j in Seg i;
    then
A2: not j in dom R2 by FINSEQ_2:124;
    not j in dom mlt(R1,R2) by A1,FINSEQ_2:124;
    then mlt(R1,R2).j = R1.j1 *0 by FUNCT_1:def 2
      .= R1.j * R2.j by A2,FUNCT_1:def 2;
    hence thesis;
  end;
  suppose
    j in Seg i;
    then j in dom mlt(R1,R2) by FINSEQ_2:124;
    hence thesis by Th17;
  end;
end;
