let f1, f2 be Function of (FreeGen p),(BoolePrime p); :: thesis: ( ( for x being Element of FreeGen p holds f1 . x = (pfexp x) | ((Seg p) /\ SetPrimes) ) & ( for x being Element of FreeGen p holds f2 . x = (pfexp x) | ((Seg p) /\ SetPrimes) ) implies f1 = f2 )
assume that
A1: for x being Element of FreeGen p holds f1 . x = (pfexp x) | ((Seg p) /\ SetPrimes) and
A2: for x being Element of FreeGen p holds f2 . x = (pfexp x) | ((Seg p) /\ SetPrimes) ; :: thesis: f1 = f2
for x being Element of FreeGen p holds f1 . x = f2 . x
proof
let x be Element of FreeGen p; :: thesis: f1 . x = f2 . x
thus f1 . x = (pfexp x) | ((Seg p) /\ SetPrimes) by A1
.= f2 . x by A2 ; :: thesis: verum
end;
hence f1 = f2 by FUNCT_2:def 8; :: thesis: verum