theorem
  a+b=1 implies a|^3+b|^3<1
proof
  assume
A1: a+b=1;
A2: 1+(a*b)*(-3)<0+1 by XREAL_1:8;
  a|^3 + b|^3=(a+b)*(a^2-a*b+b^2) by Lm6;
  then a|^3 + b|^3 = a^2+2*a*b+b^2-3*a*b by A1
    .=1^2-3*(a*b) by A1,SQUARE_1:4;
  hence thesis by A2;
end;
