
theorem qua6:
for R being Ring
for a,b,c,x being Element of R holds x * <%c,b,a%> = <%x*c,x*b,x*a%>
proof
let R be Ring; let a,b,c,x be Element of R;
set p = <%x*c,x*b,x*a%>;
H: len <%c,b,a%> <= 3 & len p <= 3 by qua2;
now let i be Element of NAT;
    i <= 2 implies i = 0 or ... or i = 2; then
    per cases;
    suppose B: i = 0;
      then <%c,b,a%>.i = c by qua1;
      hence (x*<%c,b,a%>).i = x * c by POLYNOM5:def 4 .= p.i by B,qua1;
      end;
    suppose B: i = 1;
      then <%c,b,a%>.i = b by qua1;
      hence (x*<%c,b,a%>).i = x * b by POLYNOM5:def 4 .= p.i by B,qua1;
      end;
    suppose B: i = 2;
      then <%c,b,a%>.i = a by qua1;
      hence (x*<%c,b,a%>).i = x * a by POLYNOM5:def 4 .= p.i by B,qua1;
      end;
    suppose i > 2;
      then i + 1 > 2 + 1 by XREAL_1:6;
      then C: i >= 3 by NAT_1:13;
      then <%c,b,a%>.i = 0.R by H,XXREAL_0:2,ALGSEQ_1:8;
      hence (x*<%c,b,a%>).i = x * 0.R by POLYNOM5:def 4
                           .= p.i by C,H,XXREAL_0:2,ALGSEQ_1:8;
      end;
    end;
hence thesis;
end;
