let G be non empty addMagma ; :: thesis: for g, g1, g2 being Element of G holds {g1,g2} + {g} = {(g1 + g),(g2 + g)}
let g, g1, g2 be Element of G; :: thesis: {g1,g2} + {g} = {(g1 + g),(g2 + g)}
thus {g1,g2} + {g} c= {(g1 + g),(g2 + g)} :: according to XBOOLE_0:def 10 :: thesis: {(g1 + g),(g2 + g)} c= {g1,g2} + {g}

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {g1,g2} + {g} or x in {(g1 + g),(g2 + g)} )
assume x in {g1,g2} + {g} ; :: thesis: x in {(g1 + g),(g2 + g)}
then consider h1, h2 being Element of G such that
A1: x = h1 + h2 and
A2: h1 in {g1,g2} and
A3: h2 in {g} ;
A4: ( h1 = g1 or h1 = g2 ) by A2, TARSKI:def 2;
h2 = g by A3, TARSKI:def 1;
hence x in {(g1 + g),(g2 + g)} by A1, A4, TARSKI:def 2; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(g1 + g),(g2 + g)} or x in {g1,g2} + {g} )
A5: g2 in {g1,g2} by TARSKI:def 2;
assume x in {(g1 + g),(g2 + g)} ; :: thesis: x in {g1,g2} + {g}
then A6: ( x = g1 + g or x = g2 + g ) by TARSKI:def 2;
( g in {g} & g1 in {g1,g2} ) by TARSKI:def 1, TARSKI:def 2;
hence x in {g1,g2} + {g} by A6, A5; :: thesis: verum