 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;

theorem Th3:
  sqrt <*r*> = <*sqrt r*>
proof
  set R=<*r*>;
A1: R.1=r;
A2: dom R=dom sqrt R by PARTFUN3:def 5;
  then
A3: len R=len sqrt R by FINSEQ_3:29;
  1 in Seg 1 & dom R=Seg 1 by FINSEQ_1:38;
  then len R=1 & (sqrt R).1=sqrt r by A2,A1,FINSEQ_1:40,PARTFUN3:def 5;
  hence thesis by A3,FINSEQ_1:40;
 end;
