reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  x+y=1 implies x|^3+y|^3>=1/4
proof
  assume
A1: x+y=1;
  then (x*y)*(-3)>=(1/4)*(-3) by Th18,XREAL_1:65;
  then
A2: 1+(x*y)*(-3)>=(1/4)*(-3)+1 by XREAL_1:7;
  x|^3 + y|^3=(x+y)*(x^2-x*y+y^2) by Lm6;
  then x|^3 + y|^3 = x^2+2*x*y+y^2-3*x*y by A1
    .=1^2-3*(x*y) by A1,SQUARE_1:4;
  hence thesis by A2;
end;
