let R be Relation; :: thesis: ( R is reflexive implies R ~ is reflexive )
assume R is reflexive ; :: thesis: R ~ is reflexive
then A1: R is_reflexive_in field R by Def9;

now

let x be set ; :: thesis: ( x in field (R ~) implies [x,x] in R ~ )
assume x in field (R ~) ; :: thesis: [x,x] in R ~
then x in field R by RELAT_1:21;
then [x,x] in R by A1, Def1;
hence [x,x] in R ~ by RELAT_1:def 7; :: thesis: verum

end;

then R ~ is_reflexive_in field (R ~) by Def1;
hence R ~ is reflexive by Def9; :: thesis: verum