let G be addGroup; :: thesis: for a, b, c being Element of G holds {a} * {b,c} = {(a * b),(a * c)}
let a, b, c be Element of G; :: thesis: {a} * {b,c} = {(a * b),(a * c)}
thus {a} * {b,c} c= {(a * b),(a * c)} :: according to XBOOLE_0:def 10 :: thesis: {(a * b),(a * c)} c= {a} * {b,c}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {a} * {b,c} or x in {(a * b),(a * c)} )
assume x in {a} * {b,c} ; :: thesis: x in {(a * b),(a * c)}
then consider g, h being Element of G such that
A1: x = g * h and
A2: g in {a} and
A3: h in {b,c} ;
A4: ( h = b or h = c ) by A3, TARSKI:def 2;
g = a by A2, TARSKI:def 1;
hence x in {(a * b),(a * c)} by A1, A4, TARSKI:def 2; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(a * b),(a * c)} or x in {a} * {b,c} )
A5: c in {b,c} by TARSKI:def 2;
assume x in {(a * b),(a * c)} ; :: thesis: x in {a} * {b,c}
then A6: ( x = a * b or x = a * c ) by TARSKI:def 2;
( a in {a} & b in {b,c} ) by TARSKI:def 1, TARSKI:def 2;
hence x in {a} * {b,c} by A6, A5; :: thesis: verum