let X1 be non emptyset ; :: thesis: for A1, B1 being Subset of X1 holds A1 \+\ B1 = { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) } let A1, B1 be Subset of X1; :: thesis: A1 \+\ B1 = { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) } thus
A1 \+\ B1 c= { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) } :: according to XBOOLE_0:def 10:: thesis: { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) } c= A1 \+\ B1
let a be object ; :: according to TARSKI:def 3:: thesis: ( not a in A1 \+\ B1 or a in { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) } ) assume A1:
a in A1 \+\ B1
; :: thesis: a in { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) } then reconsider x = a as Element of X1 ;
( ( x in A1 & not x in B1 ) or ( not x in A1 & x in B1 ) )
byA1, XBOOLE_0:1; hence
a in { x1 where x1 is Element of X1 : ( ( x1 in A1 & not x1 in B1 ) or ( not x1 in A1 & x1 in B1 ) ) }
; :: thesis: verum