reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem
  for f being FinSequence of REAL st len f=2 holds |.f.| = sqrt ((f.1)^2
  +(f.2)^2)
proof
  let f being FinSequence of REAL;
A1: (sqr f).1=(f.1)^2 & (sqr f).2=(f.2)^2 by VALUED_1:11;
  dom (sqr f)= dom f & Seg len (sqr f) = dom (sqr f) by FINSEQ_1:def 3
,VALUED_1:11;
  then
A2: len (sqr f) = len f by FINSEQ_1:def 3;
   reconsider f1 = (f.1)^2, f2 = (f.2)^2 as Element of REAL by XREAL_0:def 1;
  assume len f=2;
  then sqr f = <* (f.1)^2,(f.2)^2 *> by A1,A2,FINSEQ_1:44;
  then Sum sqr f = Sum (<* f1 *>^<* f2*>) by FINSEQ_1:def 9
    .=Sum <*f1*> + (f.2)^2 by RVSUM_1:74
    .= (f.1)^2 + (f.2)^2 by RVSUM_1:73;
  hence thesis;
end;
