let R be Relation; :: thesis: for X being set st R is_transitive_in X holds
R ~ is_transitive_in X
let X be set ; :: thesis: ( R is_transitive_in X implies R ~ is_transitive_in X )
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 ; :: according to RELAT_2:def 8 :: thesis: R ~ is_transitive_in X
let x, y, z be object ; :: according to RELAT_2:def 8 :: thesis: ( not x in X or not y in X or not z in X or not [x,y] in R ~ or not [y,z] in R ~ or [x,z] in R ~ )
assume that
A2: x in X and
A3: y in X and
A4: z in X and
A5: [x,y] in R ~ and
A6: [y,z] in R ~ ; :: thesis: [x,z] in R ~
A7: [z,y] in R by A6, RELAT_1:def 7;
[y,x] in R by A5, RELAT_1:def 7;
then [z,x] in R by A1, A2, A3, A4, A7;
hence [x,z] in R ~ by RELAT_1:def 7; :: thesis: verum