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 Th7:
  rseq(0,b,c,d) = b(#)rseq(0,1,c,d) &
  rseq(0,b,c,d) = (-b)(#)rseq(0,1,-c,-d)
  proof
     thus rseq(0,b,c,d) = b(#)rseq(0,1,c,d)
proof
    set f1 = rseq(0,1,c,d);
    let n be Element of NAT;
    thus rseq(0,b,c,d).n = (0*n+b)/(c*n+d) by Th5
    .= (b)*((0*n+1)/(c*n+d))
    .= b*(f1.n) by Th5
    .= (b(#)f1).n by VALUED_1:6;
  end;
     thus rseq(0,b,c,d) = (-b)(#)rseq(0,1,-c,-d)
proof
    set f1 = rseq(0,1,-c,-d);
    let n be Element of NAT;
    thus rseq(0,b,c,d).n = (0*n+b)/(c*n+d) by Th5
    .= ((-1)*b)/((-1)*(c*n+d)) by XCMPLX_1:91
    .= (-b)*((0*n+1)/((-c)*n+-d))
    .= (-b)*(f1.n) by Th5
    .= ((-b)(#)f1).n by VALUED_1:6;
  end;
end;
