reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th57:
  Sum (n-->c) = n*c
proof
  set Fn= n-->c;
  reconsider Fn as XFinSequence of COMPLEX;
A1:dom Fn = n by FUNCOP_1:13;
  now
    per cases;
    suppose
      dom Fn=0;
      hence thesis by A1;
    end;
    suppose
A2:   dom Fn>0;
      then consider f be sequence of COMPLEX such that
A3:   f.0 = Fn.0 and
A4:   for k st k+1 < len Fn holds
         f.(k + 1) = addcomplex.(f.k,Fn.(k + 1)) and
A5:   Sum Fn= f.(len Fn-1) by Def8;
      defpred P[Nat] means $1 < len Fn implies f.$1 =($1+1)*c;
A6:   for m st P[m] holds P[m+1]
      proof
        let m such that
A7:     P[m];
        assume
A8:     m + 1 < len Fn;
        then f.(m+1)=addcomplex.(f.m,Fn.(m+1)) by A4;
        then
A9:     f.(m + 1) = f.m + Fn.(m+1) by BINOP_2:def 3;
        Fn.(m+1) = c by A1,FUNCOP_1:7,A8,AFINSQ_1:86;
        hence thesis by A7,A8,A9,NAT_1:13;
      end;
      reconsider lenFn1=len Fn -1 as Element of NAT by A2,NAT_1:20;
A10:  lenFn1<lenFn1+1 by NAT_1:13;
A11:  P[0] by A3,A1,FUNCOP_1:7,AFINSQ_1:86;
      for m holds P[m] from NAT_1:sch 2(A11,A6);
      hence thesis by A5,A10,A1;
    end;
  end;
  hence thesis;
end;
