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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {a,b} |^ {c,d} or x in {(a |^ c),(a |^ d),(b |^ c),(b |^ d)} )
assume x in {a,b} |^ {c,d} ; :: thesis: x in {(a |^ c),(a |^ d),(b |^ c),(b |^ d)}
then consider g, h being Element of G such that
A1: ( x = g |^ h & g in {a,b} & h in {c,d} ) ;
( ( g = a or g = b ) & ( h = c or h = d ) ) by A1, TARSKI:def 2;
hence x in {(a |^ c),(a |^ d),(b |^ c),(b |^ d)} by A1, ENUMSET1:def 2; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {(a |^ c),(a |^ d),(b |^ c),(b |^ d)} or x in {a,b} |^ {c,d} )
assume x in {(a |^ c),(a |^ d),(b |^ c),(b |^ d)} ; :: thesis: x in {a,b} |^ {c,d}
then ( ( x = a |^ c or x = a |^ d or x = b |^ c or x = b |^ d ) & a in {a,b} & b in {a,b} & c in {c,d} & d in {c,d} ) by ENUMSET1:def 2, TARSKI:def 2;
hence x in {a,b} |^ {c,d} ; :: thesis: verum