reserve X for set;
reserve k,m,n for Nat;
reserve i for Integer;
reserve a,b,c,d,e,g,p,r,x,y for Real;
reserve z for Complex;

theorem Th8:
  rseq(a,0,c,d) = a(#)rseq(1,0,c,d) &
  rseq(a,0,c,d) = (-a)(#)rseq(1,0,-c,-d)
proof
  thus rseq(a,0,c,d) = (a)(#)rseq(1,0,c,d)
    proof
      set f = rseq(a,0,c,d);
      set f1 = rseq(1,0,c,d);
      let n be Element of NAT;
A1:   f1.n = (1*n+0)/(c*n+d) by Th5;
      thus f.n = (a*n+0)/(c*n+d) by Th5
      .= a*((1*n+0)/(c*n+d))
      .= (a(#)f1).n by A1,VALUED_1:6;
    end;
  thus rseq(a,0,c,d) = (-a)(#)rseq(1,0,-c,-d)
    proof
      set f = rseq(a,0,c,d);
      set f1 = rseq(1,0,-c,-d);
      let n be Element of NAT;
A2:   f1.n = (1*n+0)/((-c)*n+-d) by Th5;
      thus f.n = (a*n+0)/(c*n+d) by Th5
      .= ((-1)*(a*n))/((-1)*(c*n+d)) by XCMPLX_1:91
      .= (-a)*((1*n+0)/((-c)*n+-d))
      .= ((-a)(#)f1).n by A2,VALUED_1:6;
    end;
end;
