let Omega1, Omega2 be non empty set ; :: thesis: for S1 being SigmaField of Omega1
for S2 being SigmaField of Omega2
for F being random_variable of S1,S2
for y being Element of S2 st y <> {} holds
{ z where z is Element of Omega1 : F . z is Element of y } = F " y
let S1 be SigmaField of Omega1; :: thesis: for S2 being SigmaField of Omega2
for F being random_variable of S1,S2
for y being Element of S2 st y <> {} holds
{ z where z is Element of Omega1 : F . z is Element of y } = F " y
let S2 be SigmaField of Omega2; :: thesis: for F being random_variable of S1,S2
for y being Element of S2 st y <> {} holds
{ z where z is Element of Omega1 : F . z is Element of y } = F " y
let F be random_variable of S1,S2; :: thesis: for y being Element of S2 st y <> {} holds
{ z where z is Element of Omega1 : F . z is Element of y } = F " y
let y be Element of S2; :: thesis: ( y <> {} implies { z where z is Element of Omega1 : F . z is Element of y } = F " y )
assume A1: y <> {} ; :: thesis: { z where z is Element of Omega1 : F . z is Element of y } = F " y
set D = { z where z is Element of Omega1 : F . z is Element of y } ;
for x being object holds
( x in { z where z is Element of Omega1 : F . z is Element of y } iff x in F " y )

proof

let x be object ; :: thesis: ( x in { z where z is Element of Omega1 : F . z is Element of y } iff x in F " y )

hereby :: thesis: ( x in F " y implies x in { z where z is Element of Omega1 : F . z is Element of y } )

assume x in { z where z is Element of Omega1 : F . z is Element of y } ; :: thesis: x in F " y
then consider z being Element of Omega1 such that
A2: ( x = z & F . z is Element of y ) ;
z in Omega1 ;
then z in dom F by FUNCT_2:def 1;
hence x in F " y by A2, FUNCT_1:def 7, A1; :: thesis: verum

end;

assume x in F " y ; :: thesis: x in { z where z is Element of Omega1 : F . z is Element of y }
then ( x in dom F & F . x in y ) by FUNCT_1:def 7;
hence x in { z where z is Element of Omega1 : F . z is Element of y } ; :: thesis: verum

end;

hence { z where z is Element of Omega1 : F . z is Element of y } = F " y by TARSKI:2; :: thesis: verum