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

for H being Subgroup of G holds

( a in H iff H * a = carr H )

let a be Element of G; :: thesis: for H being Subgroup of G holds

( a in H iff H * a = carr H )

let H be Subgroup of G; :: thesis: ( a in H iff H * a = carr H )

thus ( a in H implies H * a = carr H ) :: thesis: ( H * a = carr H implies a in H )

( (1_ G) * a = a & 1_ G in H ) by Th46, GROUP_1:def 4;

then a in carr H by A7, Th104;

hence a in H ; :: thesis: verum

for H being Subgroup of G holds

( a in H iff H * a = carr H )

let a be Element of G; :: thesis: for H being Subgroup of G holds

( a in H iff H * a = carr H )

let H be Subgroup of G; :: thesis: ( a in H iff H * a = carr H )

thus ( a in H implies H * a = carr H ) :: thesis: ( H * a = carr H implies a in H )

proof

assume A7:
H * a = carr H
; :: thesis: a in H
assume A1:
a in H
; :: thesis: H * a = carr H

thus H * a c= carr H :: according to XBOOLE_0:def 10 :: thesis: carr H c= H * a

assume A4: x in carr H ; :: thesis: x in H * a

then A5: x in H ;

reconsider b = x as Element of G by A4;

A6: (b * (a ")) * a = b * ((a ") * a) by GROUP_1:def 3

.= b * (1_ G) by GROUP_1:def 5

.= x by GROUP_1:def 4 ;

a " in H by A1, Th51;

then b * (a ") in H by A5, Th50;

hence x in H * a by A6, Th104; :: thesis: verum

end;thus H * a c= carr H :: according to XBOOLE_0:def 10 :: thesis: carr H c= H * a

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in carr H or x in H * a )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in H * a or x in carr H )

assume x in H * a ; :: thesis: x in carr H

then consider g being Element of G such that

A2: x = g * a and

A3: g in H by Th104;

g * a in H by A1, A3, Th50;

hence x in carr H by A2; :: thesis: verum

end;assume x in H * a ; :: thesis: x in carr H

then consider g being Element of G such that

A2: x = g * a and

A3: g in H by Th104;

g * a in H by A1, A3, Th50;

hence x in carr H by A2; :: thesis: verum

assume A4: x in carr H ; :: thesis: x in H * a

then A5: x in H ;

reconsider b = x as Element of G by A4;

A6: (b * (a ")) * a = b * ((a ") * a) by GROUP_1:def 3

.= b * (1_ G) by GROUP_1:def 5

.= x by GROUP_1:def 4 ;

a " in H by A1, Th51;

then b * (a ") in H by A5, Th50;

hence x in H * a by A6, Th104; :: thesis: verum

( (1_ G) * a = a & 1_ G in H ) by Th46, GROUP_1:def 4;

then a in carr H by A7, Th104;

hence a in H ; :: thesis: verum