reserve A for non empty closed_interval Subset of REAL;

theorem
for b, c, d being Real st b < c holds
AffineMap ( d*(1 / (c - b)), d*(- b / (c - b)) ) +
AffineMap ( d*(- 1 / (c - b)), d*(c / (c - b)) ) = AffineMap (0,d)
proof
 let b, c, d be Real;
 assume A1:  b < c;
 set f = AffineMap ( d*(1 / (c - b)), d*(- b / (c - b)) );
 set g = AffineMap ( d*(- 1 / (c - b)), d*(c / (c - b)) );
 A2: dom (f+g) = dom (f) /\ dom g by VALUED_1:def 1
 .= dom (f) /\ REAL by FUNCT_2:def 1
 .= REAL /\ REAL by FUNCT_2:def 1
 .= dom AffineMap (0,d) by FUNCT_2:def 1;
 for x being object st x in dom AffineMap (0,d) holds
 (f+g).x = (AffineMap (0,d)).x
 proof
  let x be object;
  assume A3: x in dom AffineMap (0,d); then
  reconsider x as Real;
  A5:c- b <> 0 by A1;
  (f+g) . x = f.x + g.x by VALUED_1:def 1,A2,A3
  .= ((d*(1 / (c - b)))*x + d*(- b / (c - b)))
    + g.x by FCONT_1:def 4
  .= ((d*(1 / (c - b)))*x + d*(- b / (c - b)))
    + ((d*(- 1 / (c - b)))*x + d*(c / (c - b))) by FCONT_1:def 4
  .= 0*x + d* ( c / (c - b) - b / (c - b) )
  .= 0*x + d* ( (c - b) / (c - b) ) by XCMPLX_1:120
  .= 0*x + d*1 by XCMPLX_1:60,A5;
  hence thesis by FCONT_1:def 4;
 end;
 hence thesis by A2, FUNCT_1:2;
end;
