let S, X be non empty set ; :: thesis: for R being Relation of X st R is transitive holds
R is_transitive_in S
let R be Relation of X; :: thesis: ( R is transitive implies R is_transitive_in S )
assume R is transitive ; :: thesis: R is_transitive_in S
then A1: R is_transitive_in field R ;
let x, y, z be object ; :: according to RELAT_2:def 8 :: thesis: ( not x in S or not y in S or not z in S or not [x,y] in R or not [y,z] in R or [x,z] in R )
assume ( x in S & y in S & z in S ) ; :: thesis: ( not [x,y] in R or not [y,z] in R or [x,z] in R )
assume A2: [x,y] in R ; :: thesis: ( not [y,z] in R or [x,z] in R )
then A3: x in field R by RELAT_1:15;
assume A4: [y,z] in R ; :: thesis: [x,z] in R
then ( y in field R & z in field R ) by RELAT_1:15;
hence [x,z] in R by A1, A2, A4, A3; :: thesis: verum