let f1, f2 be Function of NAT,ExtREAL; :: thesis: ( ( for m being Element of NAT holds f1 . m = lim (ProjMap2 (f,m)) ) & ( for m being Element of NAT holds f2 . m = lim (ProjMap2 (f,m)) ) implies f1 = f2 )
assume that
A3: for m being Element of NAT holds f1 . m = lim (ProjMap2 (f,m)) and
A4: for m being Element of NAT holds f2 . m = lim (ProjMap2 (f,m)) ; :: thesis: f1 = f2

now :: thesis: for m being Element of NAT holds f1 . m = f2 . m

let m be Element of NAT ; :: thesis: f1 . m = f2 . m
thus f1 . m = lim (ProjMap2 (f,m)) by A3
.= f2 . m by A4 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:63; :: thesis: verum