let R be non empty RelStr ; :: thesis: for X being non empty Subset of R holds
( the carrier of (X +id ) = X & X +id is full SubRelStr of R & ( for x being Element of (X +id ) holds (X +id ) . x = x ) )
let X be non empty Subset of R; :: thesis: ( the carrier of (X +id ) = X & X +id is full SubRelStr of R & ( for x being Element of (X +id ) holds (X +id ) . x = x ) )
A1: ( RelStr(# the carrier of (X +id ),the InternalRel of (X +id ) #) = RelStr(# the carrier of ((subrelstr X) +id ),the InternalRel of ((subrelstr X) +id ) #) & the mapping of (X +id ) = (incl (subrelstr X),R) * the mapping of ((subrelstr X) +id ) ) by WAYBEL_9:def 8;
A2: the carrier of (subrelstr X) = X by YELLOW_0:def 15;
hence A3: the carrier of (X +id ) = X by A1, WAYBEL_9:def 5; :: thesis: ( X +id is full SubRelStr of R & ( for x being Element of (X +id ) holds (X +id ) . x = x ) )
A4: RelStr(# the carrier of ((subrelstr X) +id ),the InternalRel of ((subrelstr X) +id ) #) = subrelstr X by WAYBEL_9:def 5;
the InternalRel of (subrelstr X) c= the InternalRel of R by YELLOW_0:def 13;
then reconsider S = X +id as SubRelStr of R by A1, A2, A4, YELLOW_0:def 13;
S is full

proof

thus the InternalRel of S = the InternalRel of R |_2 the carrier of S by A1, A4, YELLOW_0:def 14; :: according to YELLOW_0:def 14 :: thesis: verum

end;

hence X +id is full SubRelStr of R ; :: thesis: for x being Element of (X +id ) holds (X +id ) . x = x
let x be Element of (X +id ); :: thesis: (X +id ) . x = x
id (subrelstr X) = id X by YELLOW_0:def 15;
then ( the mapping of ((subrelstr X) +id ) = id X & dom (id X) = X & incl (subrelstr X),R = id X ) by A2, Def1, RELAT_1:71, WAYBEL_9:def 5;
then the mapping of (X +id ) = id X by A1, RELAT_1:78;
hence (X +id ) . x = x by A3, FUNCT_1:35; :: thesis: verum