let r be Real; :: thesis: for X being real-membered set holds addRel (X,r) = ((curry addreal) . r) |_2 X
let X be real-membered set ; :: thesis: addRel (X,r) = ((curry addreal) . r) |_2 X
set g = (curry addreal) . r;

now :: thesis: for x, y being object holds
( ( [x,y] in addRel (X,r) implies [x,y] in ((curry addreal) . r) |_2 X ) & ( [x,y] in ((curry addreal) . r) |_2 X implies [x,y] in addRel (X,r) ) )

let x, y be object ; :: thesis: ( ( [x,y] in addRel (X,r) implies [x,y] in ((curry addreal) . r) |_2 X ) & ( [x,y] in ((curry addreal) . r) |_2 X implies [x,y] in addRel (X,r) ) )
reconsider a = x, b = y as set by TARSKI:1;

hereby :: thesis: ( [x,y] in ((curry addreal) . r) |_2 X implies [x,y] in addRel (X,r) )

assume A1: [x,y] in addRel (X,r) ; :: thesis: [x,y] in ((curry addreal) . r) |_2 X
then [a,b] in addRel (X,r) ;
then ( a in X & b in X ) by MMLQUER2:4;
then reconsider a = a, b = b as Real ;
[a,b] in addRel (X,r) by A1;
then A2: ( a in X & b in X & b = r + a ) by Th11;
( r in REAL & a in REAL ) by XREAL_0:def 1;
then [r,a] in [:REAL,REAL:] by ZFMISC_1:87;
then A3: [r,a] in dom addreal by FUNCT_2:def 1;
A4: b = addreal . (r,a) by A2, BINOP_2:def 9
.= ((curry addreal) . r) . a by A3, FUNCT_5:20 ;
a in dom ((curry addreal) . r) by A3, FUNCT_5:20;
hence [x,y] in ((curry addreal) . r) |_2 X by A2, A4, FUNCT_1:1, MMLQUER2:4; :: thesis: verum

end;

assume [x,y] in ((curry addreal) . r) |_2 X ; :: thesis: [x,y] in addRel (X,r)
then [a,b] in ((curry addreal) . r) |_2 X ;
then A5: ( a in X & b in X & [a,b] in (curry addreal) . r ) by MMLQUER2:4;
then reconsider a = a, b = b as Real ;
( r in REAL & a in REAL ) by XREAL_0:def 1;
then [r,a] in [:REAL,REAL:] by ZFMISC_1:87;
then A6: [r,a] in dom addreal by FUNCT_2:def 1;
( a in dom ((curry addreal) . r) & ((curry addreal) . r) . a = b ) by A5, FUNCT_1:1;
then b = addreal . (r,a) by A6, FUNCT_5:20
.= r + a by BINOP_2:def 9 ;
hence [x,y] in addRel (X,r) by A5, Th11; :: thesis: verum

end;

hence addRel (X,r) = ((curry addreal) . r) |_2 X by RELAT_1:def 2; :: thesis: verum