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
for n be non zero Element of NAT,
    h be PartFunc of REAL, REAL-NS n holds
 h is continuous iff
  for i be Element of NAT st i in Seg n
    holds Proj(i,n)*h is continuous
  proof
    let n be non zero Element of NAT,
    h be PartFunc of REAL, REAL-NS n;
    hereby assume
A1:   h is continuous;
      thus for i be Element of NAT st i in Seg n
      holds Proj(i,n)*h is continuous
      proof
        let i be Element of NAT;
        assume
A2:     i in Seg n;
A3:    dom Proj(i,n) = the carrier of REAL-NS n by FUNCT_2:def 1;
        rng h c= the carrier of REAL-NS n; then
A4:    dom (Proj(i,n)*h) = dom h by A3,RELAT_1:27;
        for x0 st x0 in dom (Proj(i,n)*h)
         holds Proj(i,n)*h is_continuous_in x0 by A2,Th46,A1,A4;
        hence Proj(i,n)*h is continuous;
      end;
    end;
    assume
A5: for i be Element of NAT st i in Seg n holds Proj(i,n)*h is continuous;
    let x0;
    assume A6: x0 in dom h;
    now let i be Element of NAT;
      assume A7: i in Seg n;
A8:  dom (Proj(i,n)) = the carrier of REAL-NS n by FUNCT_2:def 1;
      rng h c= the carrier of REAL-NS n; then
A9:  dom (Proj(i,n)*h) = dom h by A8,RELAT_1:27;
      Proj(i,n)*h is continuous by A5,A7;
      hence Proj(i,n)*h is_continuous_in x0 by A6,A9;
    end;
    hence h is_continuous_in x0 by Th46;
  end;
