deffunc H₁( Element of R) -> Element of bool the carrier of R = In ((Coim ( the InternalRel of R,$1)),(bool the carrier of R));
consider f being Function of the carrier of R,(bool the carrier of R) such that
A1: for x being Element of R holds f . x = H₁(x) from FUNCT_2:sch 4();
take f ; :: thesis: for x being Element of R holds f . x = Coim ( the InternalRel of R,x)
let x be Element of R; :: thesis: f . x = Coim ( the InternalRel of R,x)
A2: f . x = In ((Coim ( the InternalRel of R,x)),(bool the carrier of R)) by A1;
the InternalRel of R " {x} c= the carrier of R ;
hence f . x = Coim ( the InternalRel of R,x) by A2; :: thesis: verum