let R be Relation; for X being set st R is_transitive_in X holds
R |_2 X is transitive
let X be set ; ( R is_transitive_in X implies R |_2 X is transitive )
assume A1:
for x, y, z being object st x in X & y in X & z in X & [x,y] in R & [y,z] in R holds
[x,z] in R
; RELAT_2:def 8 R |_2 X is transitive
let x, y, z be object ; RELAT_2:def 8,RELAT_2:def 16 ( not x in field (R |_2 X) or not y in field (R |_2 X) or not z in field (R |_2 X) or not [x,y] in R |_2 X or not [y,z] in R |_2 X or [x,z] in R |_2 X )
assume that
A2:
x in field (R |_2 X)
and
A3:
y in field (R |_2 X)
and
A4:
z in field (R |_2 X)
; ( not [x,y] in R |_2 X or not [y,z] in R |_2 X or [x,z] in R |_2 X )
A5:
z in X
by A4, WELLORD1:12;
A6:
x in X
by A2, WELLORD1:12;
then A7:
[x,z] in [:X,X:]
by A5, ZFMISC_1:87;
assume that
A8:
[x,y] in R |_2 X
and
A9:
[y,z] in R |_2 X
; [x,z] in R |_2 X
A10:
[x,y] in R
by A8, XBOOLE_0:def 4;
A11:
[y,z] in R
by A9, XBOOLE_0:def 4;
y in X
by A3, WELLORD1:12;
then
[x,z] in R
by A1, A6, A5, A10, A11;
hence
[x,z] in R |_2 X
by A7, XBOOLE_0:def 4; verum