reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem
  min(min(max(f,g),max(g,h)),max(h,f)) = max(max(min(f,g),min(g,h)),min( h,f))
proof
  thus min(min(max(f,g),max(g,h)),max(h,f)) = max(min(min(max(f,g),max(g,h)),h
  ),min(min(max(f,g),max(g,h)),f)) by Th9
    .=max(min(max(f,g),min(max(g,h),h)),min(min(max(f,g),max(g,h)),f)) by Th7
    .=max(min(max(f,g),min(max(h,g),h)),min(min(max(f,g),f),max(g,h))) by Th7
    .=max(min(max(f,g),h),min(min(max(f,g),f),max(g,h))) by Th8
    .=max(min(max(f,g),h),min(f,max(g,h))) by Th8
    .=max(min(h,max(f,g)),max(min(f,g),min(f,h) )) by Th9
    .=max(max(min(f,g),min(f,h)),max(min(h,f),min(h,g))) by Th9
    .=max(max(max(min(f,g),min(f,h)),min(f,h)),min(h,g)) by Th7
    .=max(max(min(f,g),max(min(f,h),min(f,h))),min(h,g)) by Th7
    .=max(max(min(f,g),min(g,h)),min(h,f)) by Th7;
end;
