let A be set ; :: thesis: for o being BinOp of A
for e being Element of A holds
( e is_a_unity_wrt o iff for a being Element of A holds
( o . e,a = a & o . a,e = a ) )
let o be BinOp of A; :: thesis: for e being Element of A holds
( e is_a_unity_wrt o iff for a being Element of A holds
( o . e,a = a & o . a,e = a ) )
let e be Element of A; :: thesis: ( e is_a_unity_wrt o iff for a being Element of A holds
( o . e,a = a & o . a,e = a ) )
thus ( e is_a_unity_wrt o implies for a being Element of A holds
( o . e,a = a & o . a,e = a ) ) :: thesis: ( ( for a being Element of A holds
( o . e,a = a & o . a,e = a ) ) implies e is_a_unity_wrt o )

proof

assume ( e is_a_left_unity_wrt o & e is_a_right_unity_wrt o ) ; :: according to BINOP_1:def 7 :: thesis: for a being Element of A holds
( o . e,a = a & o . a,e = a )
hence for a being Element of A holds
( o . e,a = a & o . a,e = a ) by Def5, Def6; :: thesis: verum

end;

assume for a being Element of A holds
( o . e,a = a & o . a,e = a ) ; :: thesis: e is_a_unity_wrt o
hence ( ( for a being Element of A holds o . e,a = a ) & ( for a being Element of A holds o . a,e = a ) ) ; :: according to BINOP_1:def 5,BINOP_1:def 6,BINOP_1:def 7 :: thesis: verum