reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;

theorem Th5:
  |. x .| = 0 implies x = 0*n
proof
  assume
A1: |. x .| = 0;
  now
    let j;
    assume
 j in Seg n;
    reconsider r = x .j as Element of REAL by XREAL_0:def 1;
    Sum sqr x = 0 by A1,RVSUM_1:86,SQUARE_1:24;
    then Sum sqr abs x = 0 by Lm2;
    then (abs x).j = (n|->0).j by RVSUM_1:91;
    then |.r.| = (n|-> 0).j by VALUED_1:18;
    then |.r.| = 0;
    then r = 0 by ABSVALUE:2;
    hence x .j = (n|->(0 qua Real)).j;
  end;
  hence thesis by FINSEQ_2:119;
end;
