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 Th31:
  for p being Point of TOP-REAL 2 holds |.p.|<=|.p`1.|+|.p`2.|
proof
  let p be Point of TOP-REAL 2;
  (|.p.|)^2= (p`1)^2+(p`2)^2 by Th29;
  then sqrt((p`1)^2+(p`2)^2)=|.p.| by SQUARE_1:22;
  hence thesis by COMPLEX1:78;
end;
