reserve x,y,z for Real,
  a,b,c,d,e,f,g,h for Nat,
  k,l,m,n,m1,n1,m2,n2 for Integer,
  q for Rational;
reserve fs,fs1,fs2,fs3 for FinSequence;
reserve D for non empty set,
  v,v1,v2,v3 for object,
  fp for FinSequence of NAT,
  fr,fr1,fr2 for FinSequence of INT,
  ft for FinSequence of REAL;

theorem
  Del(<*v1*>,1) = {} & Del(<*v1,v2*>,1) = <*v2*> & Del(<*v1,v2*>,2) = <*
v1 *> & Del(<*v1,v2,v3*>,1) = <*v2,v3*> & Del(<*v1,v2,v3*>,2) = <*v1,v3*> & Del
  (<*v1,v2,v3*>,3) = <*v1,v2*>
proof
  thus Del(<*v1*>,1)=Del(<*v1*>^{},1) by FINSEQ_1:34
    .={} by Lm14;
  thus Del(<*v1,v2*>,1)=<*v2*> by Lm14;
  len <*v1*> =1 by FINSEQ_1:39;
  hence
A1: Del(<*v1,v2*>,2)=Del(<*v1*>^<*v2*>,len <*v1*> +1) .=<*v1*> by Lm14;
  thus Del(<*v1,v2,v3*>,1)=Del(<*v1*>^<*v2,v3*>,1) by FINSEQ_1:43
    .=<*v2,v3*> by Lm14;
A2: len<*v1,v2*>=2 by FINSEQ_1:44;
  hence Del(<*v1,v2,v3*>,2)=<*v1,v3*> by A1,Lm13;
  len <*v1,v2*>+1=3 by A2;
  hence thesis by Lm14;
end;
