let f, g be Function of (64 -tuples_on BOOLEAN),(64 -tuples_on BOOLEAN); :: thesis: ( ( for i being Element of 64 -tuples_on BOOLEAN holds f . i = DES-IP i ) & ( for i being Element of 64 -tuples_on BOOLEAN holds g . i = DES-IP i ) implies f = g )
assume A3: for i being Element of 64 -tuples_on BOOLEAN holds f . i = DES-IP i ; :: thesis: ( ex i being Element of 64 -tuples_on BOOLEAN st not g . i = DES-IP i or f = g )
assume A4: for i being Element of 64 -tuples_on BOOLEAN holds g . i = DES-IP i ; :: thesis: f = g
for i being object st i in 64 -tuples_on BOOLEAN holds
f . i = g . i

proof

let i be object ; :: thesis: ( i in 64 -tuples_on BOOLEAN implies f . i = g . i )
assume i in 64 -tuples_on BOOLEAN ; :: thesis: f . i = g . i
then reconsider i = i as Element of 64 -tuples_on BOOLEAN ;
f . i = DES-IP i by A3
.= g . i by A4 ;
hence f . i = g . i ; :: thesis: verum

end;

hence f = g ; :: thesis: verum