
theorem Th27:
  for n1,n2,n3 being Element of F_Real
  for n,u being Element of TOP-REAL 3 st n = <* n1,n2,n3 *> &
  u.3 = 1 holds |( n, u )| = n1 * u.1 + n2 * u.2 + n3
  proof
    let n1,n2,n3 be Element of F_Real;
    let n,u be Element of TOP-REAL 3;
    assume that
A1: n = <* n1,n2,n3 *> and
A2: u.3 = 1;
    n = |[n`1,n`2,n`3]| by EUCLID_5:3;
    then
A3: n`1 = n1 & n`2 = n2 & n`3 = n3 by A1,FINSEQ_1:78;
    |( n, u )| = n1 * u`1 + n2 * u`2 + n3 * u`3 by A3,EUCLID_5:29
              .= n1 * u.1 + n2 * u`2 + n3 * u`3 by EUCLID_5:def 1
              .= n1 * u.1 + n2 * u.2 + n3 * u`3 by EUCLID_5:def 2
              .= n1 * u.1 + n2 * u.2 + n3 * u.3 by EUCLID_5:def 3;
    hence thesis by A2;
  end;
