let X be non empty set ; :: thesis: for R being Relation of X st R is_transitive_in X holds
R is transitive
let R be Relation of X; :: thesis: ( R is_transitive_in X implies R is transitive )
assume A1: R is_transitive_in X ; :: thesis: R is transitive
A2: field R c= X \/ X by RELSET_1:19;
let x, y, z be set ; :: according to RELAT_2:def 8,RELAT_2:def 16 :: thesis: ( not x in field R or not y in field R or not z in field R or not [x,y] in R or not [y,z] in R or [x,z] in R )
assume A3: x in field R ; :: thesis: ( not y in field R or not z in field R or not [x,y] in R or not [y,z] in R or [x,z] in R )
assume A4: y in field R ; :: thesis: ( not z in field R or not [x,y] in R or not [y,z] in R or [x,z] in R )
assume A5: z in field R ; :: thesis: ( not [x,y] in R or not [y,z] in R or [x,z] in R )
assume ( [x,y] in R & [y,z] in R ) ; :: thesis: [x,z] in R
hence [x,z] in R by A1, A2, A3, A4, A5, RELAT_2:def 8; :: thesis: verum