theorem Th13:
  x in Seg n implies |.(0*n)+*(x,r).| = |.r.|
  proof
    set f = (0*n)+*(x,r);
A1: n in NAT by ORDINAL1:def 12;
    assume
A2: x in Seg n;
    f^2 = (0*n)+*(x,r^2) by Th12;
    then Sum (f^2) = r^2 by A2,A1,JORDAN2B:10;
    hence thesis by COMPLEX1:72;
  end;
