reserve A for non empty closed_interval Subset of REAL;

theorem Lm223:
for a,b,c,d be Real st
b > 0 & c > 0 & d > 0 & d < b holds
centroid (d(#)TrapezoidalFS (a-c,a+(d-b)/(b/c),a+(b-d)/(b/c),a+c),['a-c,a+c'])
= a
proof
 let a,b,c,d be Real;
 assume A1: b > 0 & c > 0 & d > 0 & d < b;
 A5:d-b < b-b & b-d > d-d by A1,XREAL_1:9;
 set a1 = a+(d-b)/(b/c);
 set a2 = a+(b-d)/(b/c);
 set c0 = a + c - a2;
 0+b > -d + b by XREAL_1:6,A1; then
 b/b > (b-d)/b by A1,XREAL_1:74; then
 (b/b)*c > ((b-d)/b)*c by A1,XREAL_1:68; then
 b/(b/c) > ((b-d)/b)*c by XCMPLX_1:82; then
 b/(b/c) > (b-d)/(b/c) by XCMPLX_1:82; then
 A6:b/(b/c) -(b-d)/(b/c) > (b-d)/(b/c) -(b-d)/(b/c) by XREAL_1:9;
 A3:c0 = c-(b-d)/(b/c)
 .= b/(b/c) -(b-d)/(b/c) by A1,XCMPLX_1:52;
 A4:a1-c0 = (-(b-d))/(b/c) - c + a+((b-d)/(b/c))
 .= -((b-d)/(b/c)) - c + a+((b-d)/(b/c)) by XCMPLX_1:187
 .= a-c;
 centroid ( d (#) TrapezoidalFS (a1-c0,a1,a2,a2+c0), ['a1-c0,a2+c0'])
 = (a+(-(b-d))/(b/c) + (a+(b-d)/(b/c)))/2
   by FUZZY_7:50,A1,A5,A3,A6,XREAL_1:6
 .= (a+ -((b-d)/(b/c)) + (a+(b-d)/(b/c)))/2 by XCMPLX_1:187
 .= a;
 hence thesis by A4;
end;
