let R be Relation; :: thesis: ( R is transitive iff for x, y, z being object st [x,y] in R & [y,z] in R holds
[x,z] in R )
thus ( R is transitive implies for x, y, z being object st [x,y] in R & [y,z] in R holds
[x,z] in R ) :: thesis: ( ( for x, y, z being object st [x,y] in R & [y,z] in R holds
[x,z] in R ) implies R is transitive )
proof
assume R is transitive ; :: thesis: for x, y, z being object st [x,y] in R & [y,z] in R holds
[x,z] in R
then A1: R is_transitive_in field R by RELAT_2:def 16;
let x, y, z be object ; :: thesis: ( [x,y] in R & [y,z] in R implies [x,z] in R )
assume that
A2: [x,y] in R and
A3: [y,z] in R ; :: thesis: [x,z] in R
A4: z in field R by A3, RELAT_1:15;
( x in field R & y in field R ) by A2, RELAT_1:15;
hence [x,z] in R by A1, A2, A3, A4, RELAT_2:def 8; :: thesis: verum
end;
assume for x, y, z being object st [x,y] in R & [y,z] in R holds
[x,z] in R ; :: thesis: R is transitive
then for x, y, z being object st x in field R & y in field R & z in field R & [x,y] in R & [y,z] in R holds
[x,z] in R ;
then R is_transitive_in field R by RELAT_2:def 8;
hence R is transitive by RELAT_2:def 16; :: thesis: verum