
theorem Th3:
  for n be non zero Element of NAT,
      x be Point of REAL-NS n,
      i be Nat st 1 <= i <= n holds
    ||. Proj(i,n).x .|| <= ||. x .||
proof
let n be non zero Element of NAT;
let x be Point of REAL-NS n;
let i be Nat;
assume
A1:  1 <=i & i <= n;
reconsider y = x as Element of REAL n by REAL_NS1:def 4;
A2: ||. x .|| = |.y.| by REAL_NS1:1;
Proj(i,n).x = <* proj(i,n).x *> by PDIFF_1:def 4
           .= <* y.i *> by PDIFF_1:def 1; then
A3: ||. Proj(i,n).x .|| = |.y.i.| by Th2;
reconsider p = y as Element of TOP-REAL n by EUCLID:22;
A4: i in Seg n by A1; then
A5: |.p/.i.| <= |.y.| by Th1;
i in dom y by A4,FINSEQ_1:89;
hence thesis by A2,A3,A5,PARTFUN1:def 6;
end;
