
theorem
  (for i, n being Nat st i in Seg n holds dom proj(i,n) =
REAL n & rng proj(i,n) = REAL) & for x be Real holds proj(1,1).<*x*>
  = x & proj(1,1) qua Function".x = <*x*>
proof
  now
    let i, n be Nat;
    assume
A1: i in Seg n;
    thus dom proj(i,n) = REAL n by FUNCT_2:def 1;
    proj(i,n) is onto by A1,Lm2;
    hence rng proj(i,n) = REAL by FUNCT_2:def 3;
  end;
  hence thesis by Lm1;
end;
