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 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 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)) = REAL n by FUNCT_2:def 1;
     rng h c= REAL n; then
     A4:dom (proj(i,n)*h) = dom h by A3,RELAT_1:27;
 now let x0;
  assume x0 in dom (proj(i,n)*h);
   then h is_continuous_in x0 by A1,A4;
   hence (proj(i,n)*h) is_continuous_in x0 by A2,Th43;
  end;
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)) = REAL n by FUNCT_2:def 1;
     rng h c= REAL 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 thesis by A6,Th43;
end;
