set M = { (a + b) where a, b is Element of R : ( a in I & b in J ) } ;
for y, x being Element of R st x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } holds
y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) }

proof

let y, x be Element of R; :: thesis: ( x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } implies y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } )
assume x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } ; :: thesis: y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) }
then consider a', b' being Element of R such that
A1: ( x = a' + b' & a' in I & b' in J ) ;
A2: ( y * a' in I & y * b' in J ) by A1, Def2;
(y * a') + (y * b') = y * x by A1, VECTSP_1:def 11;
hence y * x in { (a + b) where a, b is Element of R : ( a in I & b in J ) } by A2; :: thesis: verum

end;

hence I + J is left-ideal by Def2; :: thesis: verum