let f, g be Function of (REAL-NS n),(REAL-NS 1); :: thesis: ( ( for x being Point of (REAL-NS n) holds f . x = <*((proj i,n) . x)*> ) & ( for x being Point of (REAL-NS n) holds g . x = <*((proj i,n) . x)*> ) implies f = g )
assume that
A2: for x being Point of (REAL-NS n) holds f . x = <*((proj i,n) . x)*> and
A3: for x being Point of (REAL-NS n) holds g . x = <*((proj i,n) . x)*> ; :: thesis: f = g

now

let x be Point of (REAL-NS n); :: thesis: f . x = g . x
f . x = <*((proj i,n) . x)*> by A2;
hence f . x = g . x by A3; :: thesis: verum

end;

hence f = g by FUNCT_2:113; :: thesis: verum