let x be real-valued FinSequence; :: thesis: ( |(x,x)| = 0 iff x = 0* (len x) )
thus ( |(x,x)| = 0 implies x = 0* (len x) ) :: thesis: ( x = 0* (len x) implies |(x,x)| = 0 )

proof

x is FinSequence of REAL by Lm1;
then reconsider y = x as Element of REAL (len x) by EUCLID:76;
assume |(x,x)| = 0 ; :: thesis: x = 0* (len x)
then |.x.| ^2 = 0 by Th4;
then |.x.| = 0 by XCMPLX_1:6;
then y = 0* (len x) by EUCLID:8;
hence x = 0* (len x) ; :: thesis: verum

end;

assume x = 0* (len x) ; :: thesis: |(x,x)| = 0
then |.x.| = 0 by EUCLID:7;
then |(x,x)| = 0 ^2 by Th4;
hence |(x,x)| = 0 ; :: thesis: verum