set dx = Dependencies X;
set dox = Dependencies-Order X;
Dependencies X c= dom (Dependencies-Order X)

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Dependencies X or x in dom (Dependencies-Order X) )
assume x in Dependencies X ; :: thesis: x in dom (Dependencies-Order X)
then reconsider x' = x as Element of Dependencies X ;
x' <= x' ;
then [x,x] in Dependencies-Order X ;
hence x in dom (Dependencies-Order X) by RELAT_1:def 4; :: thesis: verum

end;

then A1: dom (Dependencies-Order X) = Dependencies X by XBOOLE_0:def 10;
then A2: field (Dependencies-Order X) = (Dependencies X) \/ (rng (Dependencies-Order X)) by RELAT_1:def 6
.= Dependencies X by XBOOLE_1:12 ;
thus Dependencies-Order X is total by A1, PARTFUN1:def 4; :: thesis: ( Dependencies-Order X is reflexive & Dependencies-Order X is antisymmetric & Dependencies-Order X is (F2) )
Dependencies-Order X is_reflexive_in Dependencies X

proof

let x be set ; :: according to RELAT_2:def 1 :: thesis: ( not x in Dependencies X or [x,x] in Dependencies-Order X )
assume x in Dependencies X ; :: thesis: [x,x] in Dependencies-Order X
then reconsider x' = x as Element of Dependencies X ;
x' <= x' ;
hence [x,x] in Dependencies-Order X ; :: thesis: verum

end;

hence Dependencies-Order X is reflexive by A2, RELAT_2:def 9; :: thesis: ( Dependencies-Order X is antisymmetric & Dependencies-Order X is (F2) )
Dependencies-Order X is_antisymmetric_in Dependencies X

proof

let x, y be set ; :: according to RELAT_2:def 4 :: thesis: ( not x in Dependencies X or not y in Dependencies X or not [x,y] in Dependencies-Order X or not [y,x] in Dependencies-Order X or x = y )
assume that
x in Dependencies X and
y in Dependencies X and
A3: [x,y] in Dependencies-Order X and
A4: [y,x] in Dependencies-Order X ; :: thesis: x = y
consider x', y' being Element of Dependencies X such that
A5: [x,y] = [x',y'] and
A6: x' <= y' by A3;
A7: y = y' by A5, ZFMISC_1:33;
consider y'', x'' being Element of Dependencies X such that
A8: [y,x] = [y'',x''] and
A9: y'' <= x'' by A4;
A10: x = x'' by A8, ZFMISC_1:33;
A11: y = y'' by A8, ZFMISC_1:33;
x = x' by A5, ZFMISC_1:33;
hence x = y by A6, A9, A10, A7, A11, Th13; :: thesis: verum

end;

hence Dependencies-Order X is antisymmetric by A2, RELAT_2:def 12; :: thesis: Dependencies-Order X is (F2)
Dependencies-Order X is_transitive_in Dependencies X

proof

let x, y, z be set ; :: according to RELAT_2:def 8 :: thesis: ( not x in Dependencies X or not y in Dependencies X or not z in Dependencies X or not [x,y] in Dependencies-Order X or not [y,z] in Dependencies-Order X or [x,z] in Dependencies-Order X )
assume that
x in Dependencies X and
y in Dependencies X and
z in Dependencies X and
A12: [x,y] in Dependencies-Order X and
A13: [y,z] in Dependencies-Order X ; :: thesis: [x,z] in Dependencies-Order X
consider x', y' being Element of Dependencies X such that
A14: [x,y] = [x',y'] and
A15: x' <= y' by A12;
A16: x = x' by A14, ZFMISC_1:33;
consider y'', z' being Element of Dependencies X such that
A17: [y,z] = [y'',z'] and
A18: y'' <= z' by A13;
A19: y = y'' by A17, ZFMISC_1:33;
A20: z = z' by A17, ZFMISC_1:33;
y = y' by A14, ZFMISC_1:33;
then x' <= z' by A15, A18, A19, Th14;
hence [x,z] in Dependencies-Order X by A16, A20; :: thesis: verum

end;

hence Dependencies-Order X is (F2) by A2, RELAT_2:def 16; :: thesis: verum