A6:
P is_antisymmetric_in field P
by Def12;

let a be object ; :: according to RELAT_2:def 4,RELAT_2:def 12 :: thesis: for y being object st a in field (P \ R) & y in field (P \ R) & [a,y] in P \ R & [y,a] in P \ R holds

a = y

let b be object ; :: thesis: ( a in field (P \ R) & b in field (P \ R) & [a,b] in P \ R & [b,a] in P \ R implies a = b )

assume that

( a in field (P \ R) & b in field (P \ R) ) and

A7: [a,b] in P \ R and

A8: [b,a] in P \ R ; :: thesis: a = b

A9: [b,a] in P by A8, XBOOLE_0:def 5;

A10: [a,b] in P by A7, XBOOLE_0:def 5;

then ( a in field P & b in field P ) by RELAT_1:15;

hence a = b by A6, A10, A9; :: thesis: verum

let a be object ; :: according to RELAT_2:def 4,RELAT_2:def 12 :: thesis: for y being object st a in field (P \ R) & y in field (P \ R) & [a,y] in P \ R & [y,a] in P \ R holds

a = y

let b be object ; :: thesis: ( a in field (P \ R) & b in field (P \ R) & [a,b] in P \ R & [b,a] in P \ R implies a = b )

assume that

( a in field (P \ R) & b in field (P \ R) ) and

A7: [a,b] in P \ R and

A8: [b,a] in P \ R ; :: thesis: a = b

A9: [b,a] in P by A8, XBOOLE_0:def 5;

A10: [a,b] in P by A7, XBOOLE_0:def 5;

then ( a in field P & b in field P ) by RELAT_1:15;

hence a = b by A6, A10, A9; :: thesis: verum