let G be Group; :: thesis: for H, N, N1, N2 being Subgroup of G st N = N1 /\ N2 holds

N ~ H c= (N1 ~ H) /\ (N2 ~ H)

let H be Subgroup of G; :: thesis: for N, N1, N2 being Subgroup of G st N = N1 /\ N2 holds

N ~ H c= (N1 ~ H) /\ (N2 ~ H)

let N, N1, N2 be Subgroup of G; :: thesis: ( N = N1 /\ N2 implies N ~ H c= (N1 ~ H) /\ (N2 ~ H) )

assume N = N1 /\ N2 ; :: thesis: N ~ H c= (N1 ~ H) /\ (N2 ~ H)

then A1: ( N is Subgroup of N1 & N is Subgroup of N2 ) by GROUP_2:88;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N ~ H or x in (N1 ~ H) /\ (N2 ~ H) )

assume A2: x in N ~ H ; :: thesis: x in (N1 ~ H) /\ (N2 ~ H)

( N ~ H c= N1 ~ H & N ~ H c= N2 ~ H ) by A1, Th57;

hence x in (N1 ~ H) /\ (N2 ~ H) by A2, XBOOLE_0:def 4; :: thesis: verum

N ~ H c= (N1 ~ H) /\ (N2 ~ H)

let H be Subgroup of G; :: thesis: for N, N1, N2 being Subgroup of G st N = N1 /\ N2 holds

N ~ H c= (N1 ~ H) /\ (N2 ~ H)

let N, N1, N2 be Subgroup of G; :: thesis: ( N = N1 /\ N2 implies N ~ H c= (N1 ~ H) /\ (N2 ~ H) )

assume N = N1 /\ N2 ; :: thesis: N ~ H c= (N1 ~ H) /\ (N2 ~ H)

then A1: ( N is Subgroup of N1 & N is Subgroup of N2 ) by GROUP_2:88;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N ~ H or x in (N1 ~ H) /\ (N2 ~ H) )

assume A2: x in N ~ H ; :: thesis: x in (N1 ~ H) /\ (N2 ~ H)

( N ~ H c= N1 ~ H & N ~ H c= N2 ~ H ) by A1, Th57;

hence x in (N1 ~ H) /\ (N2 ~ H) by A2, XBOOLE_0:def 4; :: thesis: verum