let X be set ; :: thesis: for R being Relation of X st R is_reflexive_in X holds
R [*] is_transitive_in X
let R be Relation of X; :: thesis: ( R is_reflexive_in X implies R [*] is_transitive_in X )
assume A1: R is_reflexive_in X ; :: thesis: R [*] is_transitive_in X
now :: thesis: 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 [*]
let x, y, z be object ; :: thesis: ( x in X & y in X & z in X & [x,y] in R [*] & [y,z] in R [*] implies [x,z] in R [*] )
assume that
A2: x in X and
y in X and
z in X and
A3: ( [x,y] in R [*] & [y,z] in R [*] ) ; :: thesis: [x,z] in R [*]
( R reduces x,y & R reduces y,z ) by A3, REWRITE1:20;
hence [x,z] in R [*] by A1, A2, Th6, REWRITE1:16; :: thesis: verum
end;
hence R [*] is_transitive_in X by RELAT_2:def 8; :: thesis: verum