reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;
reserve fr,f for FinSequence of INT;
reserve b,m for Nat;
reserve b for Integer;
reserve m for Integer;
reserve fp for FinSequence of NAT;
reserve a,m for Nat;
reserve f,g,h,k for FinSequence of REAL;

theorem Th46:
  len f = len h & len g = len k implies f^g-h^k = (f-h)^(g-k)
proof
  assume that
A1: len f = len h and
A2: len g = len k;
A3: len(f-h) = len f by A1,TOPREAL7:7;
  len(f^g) = len h + len k by A1,A2,FINSEQ_1:22;
  then len(f^g) = len(h^k) by FINSEQ_1:22;
  then
A4: len(f^g-h^k) = len(f^g) by TOPREAL7:7;
A5: len(g-k) = len g by A2,TOPREAL7:7;
  then len((f-h)^(g-k)) = len f + len g by A3,FINSEQ_1:22;
  then len(f^g-h^k) = len((f-h)^(g-k)) by A4,FINSEQ_1:22;
  then
A6: dom(f^g-h^k) = dom((f-h)^(g-k)) by FINSEQ_3:29;
  for d being Nat st d in dom((f-h)^(g-k)) holds ((f-h)^(g-k)).d = (f^g-h ^k).d
  proof
    let d be Nat;
    assume
A7: d in dom((f-h)^(g-k));
    per cases by A7,FINSEQ_1:25;
    suppose
A8:   d in dom(f-h);
      then
A9:   ((f-h)^(g-k)).d = (f-h).d by FINSEQ_1:def 7
        .= f.d - h.d by A8,VALUED_1:13;
A10:  dom f = dom(f-h) by A1,TOPREAL7:7;
A11:  dom h = dom(f-h) by A1,A3,FINSEQ_3:29;
      (f^g-h^k).d = (f^g).d - (h^k).d by A6,A8,FINSEQ_2:15,VALUED_1:13
        .= f.d - (h^k).d by A8,A10,FINSEQ_1:def 7
        .= f.d - h.d by A8,A11,FINSEQ_1:def 7;
      hence thesis by A9;
    end;
    suppose
      ex e being Nat st e in dom(g-k) & d = len(f-h) + e;
      then consider e such that
A12:  e in dom(g-k) and
A13:  d = len(f-h) + e;
      e in dom g by A2,A12,TOPREAL7:7;
      then
A14:  (f^g).d = g.e by A3,A13,FINSEQ_1:def 7;
      e in dom k by A2,A5,A12,FINSEQ_3:29;
      then
A15:  (h^k).d = k.e by A1,A3,A13,FINSEQ_1:def 7;
      ((f-h)^(g-k)).d = (g-k).e by A12,A13,FINSEQ_1:def 7
        .=g.e - k.e by A12,VALUED_1:13;
      hence thesis by A6,A12,A13,A14,A15,FINSEQ_1:28,VALUED_1:13;
    end;
  end;
  hence thesis by A6;
end;
