 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 Th2:
  sqrt (rf1^rf2) = (sqrt rf1) ^ (sqrt rf2)
proof
  set rf12=rf1^rf2;
  set s12=sqrt rf12,s1=sqrt rf1,s2=sqrt rf2;
A1: dom s12=dom rf12 by PARTFUN3:def 5;
A2: dom s2=dom rf2 by PARTFUN3:def 5;
  then
A3: len s2=len rf2 by FINSEQ_3:29;
A4: dom s1=dom rf1 by PARTFUN3:def 5;
  then
A5: len s1=len rf1 by FINSEQ_3:29;
A6: 1<=n & n<=len s12 implies s12.n=(s1^s2).n
    proof
      assume 1<=n & n<=len s12;
      then
A7:     n in dom s12 by FINSEQ_3:25;
      then
A8:     s12.n=sqrt(rf12.n) by PARTFUN3:def 5;
      per cases by A1,A7,FINSEQ_1:25;
        suppose
A9:       n in dom rf1;
          then rf12.n=rf1.n & s1.n=sqrt(rf1.n)
            by A4,FINSEQ_1:def 7,PARTFUN3:def 5;
          hence thesis by A4,A8,A9,FINSEQ_1:def 7;
        end;
        suppose ex m st m in dom rf2 & n=len rf1+m;
          then consider m such that
A10:        m in dom rf2 & n=len rf1+m;
          rf12.n=rf2.m & s2.m=sqrt(rf2.m)
            by A2,A10,FINSEQ_1:def 7,PARTFUN3:def 5;
          hence thesis by A2,A5,A8,A10,FINSEQ_1:def 7;
        end;
    end;
  len s12=len rf12 by A1,FINSEQ_3:29;
  then len s12=len s1+len s2 by A5,A3,FINSEQ_1:22;
  then len s12=len(s1^s2) by FINSEQ_1:22;
  hence thesis by A6;
end;
