reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th16:
  (R1-R2).j = R1.j - R2.j
proof
  per cases;
  suppose
A1: not j in Seg i; then
A2: not j in dom R2 by FINSEQ_2:124;
A3: not j in dom R1 by A1,FINSEQ_2:124;
    not j in dom(R1-R2) by A1,FINSEQ_2:124;
    hence (R1-R2).j = 0-0 by FUNCT_1:def 2
      .= R1.j - 0 by A3,FUNCT_1:def 2
      .= R1.j - R2.j by A2,FUNCT_1:def 2;
  end;
  suppose
    j in Seg i;
    then j in dom (R1 - R2) by FINSEQ_2:124;
    hence thesis by VALUED_1:13;
  end;
end;
