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

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

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

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

assume A1: [x,y] in multRel (X,r) ; :: thesis: [x,y] in ((curry multreal) . r) |_2 X
then [a,b] in multRel (X,r) ;
then ( a in X & b in X ) by MMLQUER2:4;
then reconsider a = a, b = b as Real ;
[a,b] in multRel (X,r) by A1;
then A2: ( a in X & b in X & b = r * a ) by Th42;
( 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 multreal by FUNCT_2:def 1;
A4: b = multreal . (r,a) by A2, BINOP_2:def 11
.= ((curry multreal) . r) . a by A3, FUNCT_5:20 ;
a in dom ((curry multreal) . r) by A3, FUNCT_5:20;
hence [x,y] in ((curry multreal) . r) |_2 X by A2, A4, FUNCT_1:1, MMLQUER2:4; :: thesis: verum

end;

assume [x,y] in ((curry multreal) . r) |_2 X ; :: thesis: [x,y] in multRel (X,r)
then [a,b] in ((curry multreal) . r) |_2 X ;
then A5: ( a in X & b in X & [a,b] in (curry multreal) . 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 multreal by FUNCT_2:def 1;
( a in dom ((curry multreal) . r) & ((curry multreal) . r) . a = b ) by A5, FUNCT_1:1;
then b = multreal . (r,a) by A6, FUNCT_5:20
.= r * a by BINOP_2:def 11 ;
hence [x,y] in multRel (X,r) by A5, Th42; :: thesis: verum

end;

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