let r, s be real number ; :: thesis: proj2 " ].r,s.[ = { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) }
set Q = proj2 " ].r,s.[;
set QQ = { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } ;

now

let z be set ; :: thesis: ( ( z in proj2 " ].r,s.[ implies z in { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } ) & ( z in { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } implies z in proj2 " ].r,s.[ ) )

hereby :: thesis: ( z in { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } implies z in proj2 " ].r,s.[ )

assume A1: z in proj2 " ].r,s.[ ; :: thesis: z in { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) }
then reconsider p = z as Point of (TOP-REAL 2) ;
proj2 . p in ].r,s.[ by A1, FUNCT_2:46;
then consider t being Real such that
A2: ( t = proj2 . p & r < t & t < s ) ;
( p `2 = proj2 . p & p = |[(p `1 ),(p `2 )]| ) by Def29, EUCLID:57;
hence z in { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } by A2; :: thesis: verum

end;

assume z in { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } ; :: thesis: z in proj2 " ].r,s.[
then consider r1, r2 being Real such that
A3: ( z = |[r1,r2]| & r < r2 & r2 < s ) ;
set p = |[r1,r2]|;
A4: proj2 . |[r1,r2]| = |[r1,r2]| `2 by Def29
.= r2 by EUCLID:56 ;
r2 in ].r,s.[ by A3;
hence z in proj2 " ].r,s.[ by A3, A4, FUNCT_2:46; :: thesis: verum

end;

hence proj2 " ].r,s.[ = { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) } by TARSKI:2; :: thesis: verum