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
  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;
