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 Th11:
for x be Element of REAL,i be Nat st
 1 <= i & i <= m & x <> 0 holds Replace(0*m,i,x) <> 0*m
proof
   let x be Element of REAL;
   let i be Nat;
   assume A1: 1 <= i & i <= m & x <> 0;
then A2:i in Seg m;
   assume A3: Replace((0*m),i,x) = 0*m;
    len(0*m) = m by CARD_1:def 7;
   then Seg m = proj1(0*m) by FINSEQ_1:def 3;
   then x = (0*m).i by A3,A2,FUNCT_7:31;
   hence contradiction by A1;
end;
