reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th18:
for x be Point of REAL-NS 1, i be Nat st
  1 <=i & i <= m & x <> 0.(REAL-NS 1) holds
    reproj(i,0.(REAL-NS m)).x <> 0.(REAL-NS m)
proof
   let x be Point of REAL-NS 1, i be Nat;
   assume A1: 1 <=i & i <= m & x <> 0.(REAL-NS 1);
   consider q1 be Element of REAL, z1 be Element of REAL m such that
A2: x = <*q1*> & z1 = 0.(REAL-NS m)
    & reproj(i,0.(REAL-NS m)).x = reproj(i,z1).q1 by PDIFF_1:def 6;
A3:0.(REAL-NS m) = 0*m by REAL_NS1:def 4;
    now assume q1=0;
    then <*q1*> = 0*1 by FINSEQ_2:59;
    hence contradiction by A2,A1,REAL_NS1:def 4;
   end;
   hence thesis by A2,A3,A1,Th17;
end;
