reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th24:
  for r be Real,
    f,g be real-valued FinSequence holds
     r |^ (f^g) = (r |^ f) ^ (r |^ g)
proof
  let r be Real,f,g be real-valued FinSequence;
  set fg=f^g,rf=r|^f,rg=r|^g;
  A1:len fg=len f+len g & len (rf^rg) = len rf + len rg by FINSEQ_1:22;
  A2:len rf = len f & len rg = len g & len (r|^fg) = len fg by CARD_1:def 7;
  then A3:dom f = dom rf & dom g = dom rg by FINSEQ_3:29;
  for i st 1<= i & i <= len fg holds (r|^fg).i = (rf^rg).i
  proof
    let i;
    assume 1<= i & i <= len fg;
    then A4:i in dom fg by FINSEQ_3:25;
    then A5: (r|^fg).i = r to_power (fg.i) by Def4;
    per cases by A4,FINSEQ_1:25;
    suppose A6: i in dom f;
      then fg.i=f.i & (rf^rg).i = rf.i by A3,FINSEQ_1:def 7;
      hence thesis by A6,Def4,A5;
    end;
    suppose ex j st j in dom g & i=len f+j;
      then consider j such that
      A7:j in dom g & i = len f+j;
      fg.i = g.j & (rf^rg).i = rg.j by A3,A7,A2,FINSEQ_1:def 7;
      hence thesis by A7,Def4,A5;
    end;
  end;
  hence thesis by A1,A2;
end;
