let G be Group; :: thesis: for A, B being non empty Subset of G
for N being Subgroup of G
for x being Element of G st x in N ~ (A * B) holds
x * N meets A * B
let A, B be non empty Subset of G; :: thesis: for N being Subgroup of G
for x being Element of G st x in N ~ (A * B) holds
x * N meets A * B
let N be Subgroup of G; :: thesis: for x being Element of G st x in N ~ (A * B) holds
x * N meets A * B
let x be Element of G; :: thesis: ( x in N ~ (A * B) implies x * N meets A * B )
assume x in N ~ (A * B) ; :: thesis: x * N meets A * B
then consider x1 being Element of G such that
A1: ( x = x1 & x1 * N meets A * B ) ;
thus x * N meets A * B by A1; :: thesis: verum