let G be Group; :: thesis: for a being Element of G

for H1, H2 being Subgroup of G holds a * (H1 /\ H2) = (a * H1) /\ (a * H2)

let a be Element of G; :: thesis: for H1, H2 being Subgroup of G holds a * (H1 /\ H2) = (a * H1) /\ (a * H2)

let H1, H2 be Subgroup of G; :: thesis: a * (H1 /\ H2) = (a * H1) /\ (a * H2)

thus a * (H1 /\ H2) c= (a * H1) /\ (a * H2) :: according to XBOOLE_0:def 10 :: thesis: (a * H1) /\ (a * H2) c= a * (H1 /\ H2)

assume A4: x in (a * H1) /\ (a * H2) ; :: thesis: x in a * (H1 /\ H2)

then x in a * H1 by XBOOLE_0:def 4;

then consider g being Element of G such that

A5: x = a * g and

A6: g in H1 by Th103;

x in a * H2 by A4, XBOOLE_0:def 4;

then consider g1 being Element of G such that

A7: x = a * g1 and

A8: g1 in H2 by Th103;

g = g1 by A5, A7, GROUP_1:6;

then g in H1 /\ H2 by A6, A8, Th82;

hence x in a * (H1 /\ H2) by A5, Th103; :: thesis: verum

for H1, H2 being Subgroup of G holds a * (H1 /\ H2) = (a * H1) /\ (a * H2)

let a be Element of G; :: thesis: for H1, H2 being Subgroup of G holds a * (H1 /\ H2) = (a * H1) /\ (a * H2)

let H1, H2 be Subgroup of G; :: thesis: a * (H1 /\ H2) = (a * H1) /\ (a * H2)

thus a * (H1 /\ H2) c= (a * H1) /\ (a * H2) :: according to XBOOLE_0:def 10 :: thesis: (a * H1) /\ (a * H2) c= a * (H1 /\ H2)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (a * H1) /\ (a * H2) or x in a * (H1 /\ H2) )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in a * (H1 /\ H2) or x in (a * H1) /\ (a * H2) )

assume x in a * (H1 /\ H2) ; :: thesis: x in (a * H1) /\ (a * H2)

then consider g being Element of G such that

A1: x = a * g and

A2: g in H1 /\ H2 by Th103;

g in H2 by A2, Th82;

then A3: x in a * H2 by A1, Th103;

g in H1 by A2, Th82;

then x in a * H1 by A1, Th103;

hence x in (a * H1) /\ (a * H2) by A3, XBOOLE_0:def 4; :: thesis: verum

end;assume x in a * (H1 /\ H2) ; :: thesis: x in (a * H1) /\ (a * H2)

then consider g being Element of G such that

A1: x = a * g and

A2: g in H1 /\ H2 by Th103;

g in H2 by A2, Th82;

then A3: x in a * H2 by A1, Th103;

g in H1 by A2, Th82;

then x in a * H1 by A1, Th103;

hence x in (a * H1) /\ (a * H2) by A3, XBOOLE_0:def 4; :: thesis: verum

assume A4: x in (a * H1) /\ (a * H2) ; :: thesis: x in a * (H1 /\ H2)

then x in a * H1 by XBOOLE_0:def 4;

then consider g being Element of G such that

A5: x = a * g and

A6: g in H1 by Th103;

x in a * H2 by A4, XBOOLE_0:def 4;

then consider g1 being Element of G such that

A7: x = a * g1 and

A8: g1 in H2 by Th103;

g = g1 by A5, A7, GROUP_1:6;

then g in H1 /\ H2 by A6, A8, Th82;

hence x in a * (H1 /\ H2) by A5, Th103; :: thesis: verum