let G be non empty addMagma ; :: thesis: for g, g1, g2, h being Element of G holds {g,h} + {g1,g2} = {(g + g1),(g + g2),(h + g1),(h + g2)}
let g, g1, g2, h be Element of G; :: thesis: {g,h} + {g1,g2} = {(g + g1),(g + g2),(h + g1),(h + g2)}
set A = {g,h} + {g1,g2};
set B = {(g + g1),(g + g2),(h + g1),(h + g2)};
thus {g,h} + {g1,g2} c= {(g + g1),(g + g2),(h + g1),(h + g2)} :: according to XBOOLE_0:def 10 :: thesis: {(g + g1),(g + g2),(h + g1),(h + g2)} c= {g,h} + {g1,g2}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {g,h} + {g1,g2} or x in {(g + g1),(g + g2),(h + g1),(h + g2)} )
assume x in {g,h} + {g1,g2} ; :: thesis: x in {(g + g1),(g + g2),(h + g1),(h + g2)}
then consider h1, h2 being Element of G such that
A1: x = h1 + h2 and
A2: h1 in {g,h} and
A3: h2 in {g1,g2} ;
A4: ( h2 = g1 or h2 = g2 ) by A3, TARSKI:def 2;
( h1 = g or h1 = h ) by A2, TARSKI:def 2;
hence x in {(g + g1),(g + g2),(h + g1),(h + g2)} by A1, A4, ENUMSET1:def 2; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {(g + g1),(g + g2),(h + g1),(h + g2)} or x in {g,h} + {g1,g2} )
A5: ( g1 in {g1,g2} & g2 in {g1,g2} ) by TARSKI:def 2;
assume x in {(g + g1),(g + g2),(h + g1),(h + g2)} ; :: thesis: x in {g,h} + {g1,g2}
then A6: ( x = g + g1 or x = g + g2 or x = h + g1 or x = h + g2 ) by ENUMSET1:def 2;
( g in {g,h} & h in {g,h} ) by TARSKI:def 2;
hence x in {g,h} + {g1,g2} by A6, A5; :: thesis: verum