let f1, f2 be Function of NAT,ExtREAL; :: thesis: ( ( for n being Element of NAT holds f1 . n = lim (ProjMap1 (f,n)) ) & ( for n being Element of NAT holds f2 . n = lim (ProjMap1 (f,n)) ) implies f1 = f2 )
assume that
A3: for n being Element of NAT holds f1 . n = lim (ProjMap1 (f,n)) and
A4: for n being Element of NAT holds f2 . n = lim (ProjMap1 (f,n)) ; :: thesis: f1 = f2

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

let n be Element of NAT ; :: thesis: f1 . n = f2 . n
thus f1 . n = lim (ProjMap1 (f,n)) by A3
.= f2 . n by A4 ; :: thesis: verum

end;

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