let G be Group; :: thesis: Left_Cosets ((1). G) = { {a} where a is Element of G : verum }
set A = { {a} where a is Element of G : verum } ;
thus Left_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= Left_Cosets ((1). G)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Left_Cosets ((1). G) or x in { {a} where a is Element of G : verum } )
assume A1: x in Left_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 = g * ((1). G) by ;
x = {g} by ;
hence x in { {a} where a is Element of G : verum } ; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { {a} where a is Element of G : verum } or x in Left_Cosets ((1). G) )
assume x in { {a} where a is Element of G : verum } ; :: thesis: x in Left_Cosets ((1). G)
then consider a being Element of G such that
A3: x = {a} ;
a * ((1). G) = {a} by Th110;
hence x in Left_Cosets ((1). G) by ; :: thesis: verum