let f1, f2 be UnOp of G; :: thesis: ( ( for a being Element of G holds f1 . a = Bottom a ) & ( for a being Element of G holds f2 . a = Bottom a ) implies f1 = f2 )

assume that

A2: for a being Element of G holds f1 . a = Bottom a and

A3: for a being Element of G holds f2 . a = Bottom a ; :: thesis: f1 = f2

assume that

A2: for a being Element of G holds f1 . a = Bottom a and

A3: for a being Element of G holds f2 . a = Bottom a ; :: thesis: f1 = f2

now :: thesis: for a being Element of G holds f1 . a = f2 . a

hence
f1 = f2
by FUNCT_2:63; :: thesis: verumlet a be Element of G; :: thesis: f1 . a = f2 . a

thus f1 . a = Bottom a by A2

.= f2 . a by A3 ; :: thesis: verum

end;thus f1 . a = Bottom a by A2

.= f2 . a by A3 ; :: thesis: verum