let G be Group; :: thesis: Right_Cosets ((1). G) = { {a} where a is Element of G : verum }

set A = { {a} where a is Element of G : verum } ;

thus Right_Cosets ((1). G) c= { {a} where a is Element of G : verum } :: according to XBOOLE_0:def 10 :: thesis: { {a} where a is Element of G : verum } c= Right_Cosets ((1). G)

assume x in { {a} where a is Element of G : verum } ; :: thesis: x in Right_Cosets ((1). G)

then consider a being Element of G such that

A3: x = {a} ;

((1). G) * a = {a} by Th110;

hence x in Right_Cosets ((1). G) by A3, Def16; :: thesis: verum

set A = { {a} where a is Element of G : verum } ;

thus Right_Cosets ((1). G) c= { {a} where a is Element of G : verum } :: according to XBOOLE_0:def 10 :: thesis: { {a} where a is Element of G : verum } c= Right_Cosets ((1). G)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { {a} where a is Element of G : verum } or x in Right_Cosets ((1). G) )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Right_Cosets ((1). G) or x in { {a} where a is Element of G : verum } )

assume A1: x in Right_Cosets ((1). G) ; :: thesis: x in { {a} where a is Element of G : verum }

then reconsider X = x as Subset of G ;

consider g being Element of G such that

A2: X = ((1). G) * g by A1, Def16;

x = {g} by A2, Th110;

hence x in { {a} where a is Element of G : verum } ; :: thesis: verum

end;assume A1: x in Right_Cosets ((1). G) ; :: thesis: x in { {a} where a is Element of G : verum }

then reconsider X = x as Subset of G ;

consider g being Element of G such that

A2: X = ((1). G) * g by A1, Def16;

x = {g} by A2, Th110;

hence x in { {a} where a is Element of G : verum } ; :: thesis: verum

assume x in { {a} where a is Element of G : verum } ; :: thesis: x in Right_Cosets ((1). G)

then consider a being Element of G such that

A3: x = {a} ;

((1). G) * a = {a} by Th110;

hence x in Right_Cosets ((1). G) by A3, Def16; :: thesis: verum