reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th6:
  for x,y be Real, u,v being Element of REAL 2
   st u = <*x,y*> & v = <*y,x*>
  holds |.u.| = |.v.|
  proof
    let x,y be Real, u,v be Element of REAL 2;
    assume
    A1: u = <*x,y*> & v = <*y,x*>; then
    A3: len v = 2 & v.1 = y & v.2 = x by FINSEQ_1:44;
    len u = 2 & u.1 = x & u.2 = y by A1,FINSEQ_1:44;

    hence |.u.| = sqrt (x^2 + y^2) by EUCLID_3:22
              .= |.v.| by A3,EUCLID_3:22;
  end;
