let T be non empty RelStr ; :: thesis: for p being Element of T
for k being Element of NAT st T is filled holds
U_FT (p,0) c= U_FT (p,k)
let p be Element of T; :: thesis: for k being Element of NAT st T is filled holds
U_FT (p,0) c= U_FT (p,k)
let k be Element of NAT ; :: thesis: ( T is filled implies U_FT (p,0) c= U_FT (p,k) )
defpred S₁[ Element of NAT ] means U_FT (p,0) c= U_FT (p,$1);
assume A1: T is filled ; :: thesis: U_FT (p,0) c= U_FT (p,k)
A2: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
U_FT (p,k) c= U_FT (p,(k + 1)) by A1, Th7;
hence S₁[k + 1] by A3, XBOOLE_1:1; :: thesis: verum

end;

A4: S₁[ 0 ] ;
for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A4, A2);
hence U_FT (p,0) c= U_FT (p,k) ; :: thesis: verum