reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th61:
  id [#](REAL) is_convex_on REAL
proof
  set i = id [#](REAL);
  thus REAL c= dom i by FUNCT_1:17;
  let p be Real;
  assume that
  0<=p and
  p<=1;
  let x,y be Real;
  assume that
A1: x in REAL and
A2: y in REAL and
A3: p*x+(1-p)*y in REAL;
  i.x = x & i.y = y by A1,A2,FUNCT_1:17;
  hence thesis by A3,FUNCT_1:17;
end;
