let R be Relation; :: thesis: ( R is transitive implies R ~ is transitive )
assume R is transitive ; :: thesis: R ~ is transitive
then A1: R is_transitive_in field R by Def16;

now

let x, y, z be set ; :: thesis: ( x in field (R ~) & y in field (R ~) & z in field (R ~) & [x,y] in R ~ & [y,z] in R ~ implies [x,z] in R ~ )
assume that
A2: ( x in field (R ~) & y in field (R ~) ) and
A3: z in field (R ~) and
A4: [x,y] in R ~ and
A5: [y,z] in R ~ ; :: thesis: [x,z] in R ~
A6: ( x in field R & y in field R ) by A2, RELAT_1:21;
A7: [y,x] in R by A4, RELAT_1:def 7;
( z in field R & [z,y] in R ) by A3, A5, RELAT_1:21, RELAT_1:def 7;
then [z,x] in R by A1, A6, A7, Def8;
hence [x,z] in R ~ by RELAT_1:def 7; :: thesis: verum

end;

then R ~ is_transitive_in field (R ~) by Def8;
hence R ~ is transitive by Def16; :: thesis: verum