reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

theorem Th45:
  for x0 be Point of REAL-NS n st 1 <= i & i <= n holds
  Proj(i,n) is_continuous_in x0
proof
let x0 be Point of REAL-NS n;
assume A1: 1 <= i & i <= n;
A2: dom (Proj(i,n)) = the carrier of REAL-NS n by FUNCT_2:def 1;
now let r be Real;
   assume A3: 0<r;
   take s = r;
   thus 0< s by A3;
    now let x1 be Point of REAL-NS n;
      assume A4: x1 in dom (Proj(i,n)) & ||.x1-x0.|| < s;
      Proj(i,n)/.(x1-x0) = Proj(i,n)/.x1-Proj(i,n)/.x0 by A1,PDIFF_8:11;
      then ||. Proj(i,n)/.x1-Proj(i,n)/.x0.|| <= ||. x1-x0 .||
        by A1,PDIFF_8:3;
      hence ||. Proj(i,n)/.x1-Proj(i,n)/.x0.|| < r by A4,XXREAL_0:2;
    end;
    hence for x1 be Point of REAL-NS n st x1 in dom (Proj(i,n))
     & ||.x1-x0.|| < s holds ||. Proj(i,n)/.x1-Proj(i,n)/.x0 .||<r;
 end;
 hence thesis by A2,NFCONT_1:7;
end;
