reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;
reserve p,p1,p2 for Point of TOP-REAL N;

theorem Th29:
  for p being Point of TOP-REAL 2 holds |.p.|^2 = (p`1)^2+(p`2)^2
proof
  let p be Point of TOP-REAL 2;
  reconsider w=p as Element of REAL 2 by EUCLID:22;
A1: (sqr w).2=(p`2)^2 by VALUED_1:11;
  0 <= Sum sqr w by RVSUM_1:86;
  then
A2: |.p.|^2=Sum sqr w by SQUARE_1:def 2;
  len sqr w =2 & (sqr w).1=(p`1)^2 by CARD_1:def 7,VALUED_1:11;
  then sqr w=<*(p`1)^2,(p`2)^2*> by A1,FINSEQ_1:44;
  hence thesis by A2,RVSUM_1:77;
end;
